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Published online by Cambridge University Press:  03 November 2022

Rupert L. Frank ,
Ari Laptev  and
Timo Weidl
Rupert L. Frank
Affiliation:
Ludwig-Maximilians-Universität München
Ari Laptev
Affiliation:
Imperial College of Science, Technology and Medicine, London
Timo Weidl
Affiliation:
Universität Stuttgart
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Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities , pp. 471 - 503
Publisher: Cambridge University Press
Print publication year: 2022

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References

Abramov, A. A., Aslanyan, A., and Davies, E. B. 2001. Bounds on complex eigenvalues and resonances. J. Phys. A, 34(1), 57–72.Google Scholar
Abramowitz, M., and Stegun, I. A. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55.Google Scholar
Adams, D. R., and Hedberg, L. I. 1996. Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Adams, R. A., and Fournier, J. J. F. 2003. Sobolev Spaces. Second edn. Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier/Academic Press, Amsterdam.Google Scholar
Agmon, S. 1965. On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems. Comm. Pure Appl. Math., 18, 627–663.CrossRefGoogle Scholar
Agmon, S. 1967/68. Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators. Arch. Rational Mech. Anal., 28, 165–183.CrossRefGoogle Scholar
Agmon, S., and Kannai, Y. 1967. On the asymptotic behavoir of spectral functions and resolvant kernels of elliptic operators. Israel J. Math., 5, 1–30.Google Scholar
Agranovič, Z. S., and Marčenko, V. A. 1960. Reconstruction of the potential energy from the dispersion matrix. Amer. Math. Soc. Transl. (2), 16, 355–357.Google Scholar
Agranovich, Z. S., and Marchenko, V. A. 1963. The Inverse Problem of Scattering Theory. Translated from the Russian by Seckler, B. D.. Gordon and Breach Science Publishers, New York and London.Google Scholar
Aizenman, M., and Lieb, E. H. 1978. On semiclassical bounds for eigenvalues of Schrödinger operators. Phys. Lett. A, 66(6), 427–429.Google Scholar
Aizenman, M., and Simon, B. 1982. Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math., 35(2), 209–273.CrossRefGoogle Scholar
Aizenman, M., Elgart, A., Naboko, S., Schenker, J. H., and Stolz, G. 2006. Moment analysis for localization in random Schrödinger operators. Invent. Math., 163(2), 343–413.Google Scholar
Akhiezer, N. I., and Glazman, I. M. 1963. Theory of Linear Operators in Hilbert Space, in two volumes. English translation, with a preface by Nestell, M., 1993. Two volumes bound as one. Dover Publications, Inc., New York.Google Scholar
Alama, S., Deift, P. A., and Hempel, R. 1989. Eigenvalue branches of the Schrödinger operator H - λW in a gap of σ(H ). Comm. Math. Phys., 121(2), 291–321.Google Scholar
Allen, M., Kriventsov, D., and Neumayer, R. 2021. Sharp quantitative Faber–Krahn inequalities and the Alt–Caffarelli–Friedman monotonicity formula. Preprint, available at ArXiv:2107.03505Google Scholar
Alonso, A., and Simon, B. 1980. The Birman–Kreĭn–Vishik theory of selfadjoint extensions of semibounded operators. J. Operator Theory, 4(2), 251–270.Google Scholar
Alvino, A. 1977. Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. A (5), 14(1), 148–156.Google Scholar
Andrews, B., and Clutterbuck, J. 2011. Proof of the fundamental gap conjecture. J. Amer. Math. Soc., 24(3), 899–916.CrossRefGoogle Scholar
Araki, H. 1990. On an inequality of Lieb and Thirring. Lett. Math. Phys., 19(2), 167–170.Google Scholar
Arazy, J., and Zelenko, L. 2006. Virtual eigenvalues of the high order Schrödinger operator. I. Integral Equations Operator Theory, 55(2), 189–231.Google Scholar
Ashbaugh, M. S., and Benguria, R. D. 1992. A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. of Math. (2), 135(3), 601–628.Google Scholar
Ashbaugh, M. S., and Benguria, R. D. 2007. Isoperimetric inequalities for eigenvalues of the Laplacian. Pages 105–139 of: Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday. Gesztesy, F., Deift, P., Galvez, C., Perry, P., and Schlag, W. (editors). Proc. Sympos. Pure Math., vol. 76. Amer. Math. Soc., Providence, RI.CrossRefGoogle Scholar
Ashbaugh, M. S., Gesztesy, F., Mitrea, M., Shterenberg, R., and Teschl, G. 2010a. The Krein–von Neumann extension and its connection to an abstract buckling problem. Math. Nachr., 283(2), 165–179.CrossRefGoogle Scholar
Ashbaugh, M. S., Gesztesy, F., Mitrea, M., and Teschl, G. 2010b. Spectral theory for perturbed Krein Laplacians in nonsmooth domains. Adv. Math., 223(4), 1372–1467.Google Scholar
Ashu, A. M. 2013. Some properties of Bessel functions with applications to Neumann eigenvalues in the unit disc. Student Paper, Lund University. https://lup.lub.lu.se/student-papers/search/publication/7370411.Google Scholar
Aubin, T.. 1976. Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geometry, 11(4), 573–598.CrossRefGoogle Scholar
Aubin, T.. 1998. Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin.Google Scholar
Avakumović, Vojislav G. 1956. Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z., 65, 327–344.Google Scholar
Aviles, P.. 1986. Symmetry theorems related to Pompeiu’s problem. Amer. J. Math., 108(5), 1023–1036.Google Scholar
Avron, J., Herbst, I., and Simon, B. 1978. Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J., 45(4), 847–883.Google Scholar
Balinsky, A. A., Evans, W. D., and Lewis, R. T. 2015. The Analysis and Geometry of Hardy’s Inequality. Universitext. Springer, Cham.CrossRefGoogle Scholar
Bandle, C.. 1980. Isoperimetric Inequalities and Applications. Monographs and Studies in Mathematics, vol. 7. Pitman (Advanced Publishing Program), Boston, MA and London.Google Scholar
Bargmann, V. 1952. On the number of bound states in a central field of force. Proc. Nat. Acad. Sci. U.S.A., 38, 961–966.CrossRefGoogle Scholar
Barsegyan, D. S. 2007. On inequalities of Lieb–Thirring type. Mat. Zametki, 82(4), 504–514. English translation in Math. Notes 82 (2007), no. 3–4, 451–460.Google Scholar
Barseghyan, D., Exner, P., Kovařík, H., and Weidl, T. 2016. Semiclassical bounds in magnetic bottles. Rev. Math. Phys., 28(1), 1650002, 29.Google Scholar
Beals, R. 1967. Classes of compact operators and eigenvalue distributions for elliptic operators. Amer. J. Math., 89, 1056–1072.CrossRefGoogle Scholar
Bebendorf, M. 2003. A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen, 22(4), 751–756.CrossRefGoogle Scholar
Beckner, W. 2004. Estimates on Moser embedding. Potential Anal., 20(4), 345–359.CrossRefGoogle Scholar
Benguria, R. D. 2011. Isoperimetric inequalities for eigenvalues of the Laplacian. Pages 21–60 of: Entropy and the Quantum II. Contemp. Math., vol. 552. Amer. Math. Soc., Providence, RI.CrossRefGoogle Scholar
Benguria, R., and Loss, M. 2000. A simple proof of a theorem of Laptev and Weidl. Math. Res. Lett., 7(2–3), 195–203.CrossRefGoogle Scholar
Benguria, R. D., and Loss, M. 2004. Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane. Pages 53–61 of: Partial Differential Equations and Inverse Problems. Conca, C., Manásevich, R., Uhlmann, G., and Vogelius, M. S. (editors). Contemp. Math., vol. 362. Amer. Math. Soc., Providence, RI.Google Scholar
Benguria, R. D., Linde, H., and Loewe, B. 2012. Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator. Bull. Math. Sci., 2(1), 1–56.Google Scholar
Bérard, P. H. 1977. On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z., 155(3), 249–276.Google Scholar
Bérard, P. H. 1980. Spectres et groupes cristallographiques. I. Domaines euclidiens. Invent. Math., 58(2), 179–199.Google Scholar
Bérard, P. H. 1986. Spectral Geometry: Direct and Inverse Problems. With appendices by Besson, G., and by Bérard, P. and Berger, M.. Lecture Notes in Mathematics, vol. 1207. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Berezin, F. A. 1972a. Convex functions of operators. Mat. Sb. (N.S.), 88(130), 268–276. English translation in Math. USSR-Sb., 17(2), (1972), 269–277.Google Scholar
Berezin, F. A. 1972b. Covariant and contravariant symbols of operators. Izv. Akad. Nauk SSSR Ser. Mat., 36, 1134–1167. English translation in Math. USSR-Izv. 6, (1972), 1117–1151.Google Scholar
Berry, M. V. 1980. Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals. Pages 13–28 of: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979). Osserman, R. and Weinstein, A. (editors). Proc. Sympos. Pure Math., XXXVI. Amer. Math. Soc., Providence, RI.Google Scholar
Bez, N., Hong, Y., Lee, S., Nakamura, S., and Sawano, Y. 2019. On the Strichartz estimates for orthonormal systems of initial data with regularity. Adv. Math., 354, 106736, 37.CrossRefGoogle Scholar
Bez, N., Lee, S., and Nakamura, S. 2020. Maximal estimates for the Schrödinger equation with orthonormal initial data. Selecta Math. (N.S.), 26(4), Paper No. 52, 24.CrossRefGoogle Scholar
Bez, N., Lee, S., and Nakamura, S. 2021. Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates. Forum Math. Sigma, 9, Paper No. e1, 52.Google Scholar
Bianchi, G., and Egnell, H. 1991. A note on the Sobolev inequality. J. Funct. Anal., 100(1), 18–24.CrossRefGoogle Scholar
Birman, M. Š. 1959. Perturbations of quadratic forms and the spectrum of singular boundary value problems. Dokl. Akad. Nauk SSSR, 125, 471–474.Google Scholar
Birman, M. Š. 1961. On the spectrum of singular boundary-value problems. Mat. Sb. (N.S.), 55 (97), 125–174. English translation: Pages 23–80 of: Eleven Papers on Analysis, Amer. Math. Soc. Transl. 53, (1966). Amer. Math. Soc., Providence, RI.Google Scholar
Birman, M. Š., and Borzov, V. V. 1971. The asymptotic behavior of the discrete spectrum of certain singular differential operators. Pages 24–28 of: Spectral Theory. Birman, M.Sh. (editor). Problems of Mathematical Physics, No. 5. Izdat. Leningrad. Univ., Leningrad. English translation in Spectral Theory, Topics in Mathematical Physics 5, 1–18 (1972). Consultants Bureau, New York and London.Google Scholar
Birman, M. Sh., and Laptev, A. 1996. The negative discrete spectrum of a two-dimensional Schrödinger operator. Comm. Pure Appl. Math., 49(9), 967–997.Google Scholar
Birman, M.Š., and Solomjak, M.Z. 1966. Approximation of functions of the Wpα-classes by piece-wise-polynomial functions. Dokl. Akad. Nauk SSSR, 171, 1015–1018. English translation in Sov. Math. Dokl., 7, 1573-1577, 1966. Corrections ibid. 8(1), vi, (1967).Google Scholar
Birman, M. Š., and Solomjak, M. Z. 1967. Piecewise polynomial approximations of functions of classes Wpα . Mat. Sb. (N.S.), 73 (115), 331–355. English translation in Math USSR Sb., 2 (1967), 295–317.Google Scholar
Birman, M. Š., and Solomjak, M. Z. 1970. The principal term of the spectral asymptotics for “non-smooth” elliptic problems. Funkcional. Anal. i Priložen., 4(4), 1–13. English translation in Functional Anal. Appl., 4 (1970), 265–275.Google Scholar
Birman, M. Š., and Solomjak, M. Z. 1971. The asymptotics of the spectrum of “non-smooth” elliptic equations. Funkcional. Anal. i Priložen., 5(1), 69–70. English translation in Functional Anal. Appl., 5, (1971), 56–57.Google Scholar
Birman, M. Š., and Solomjak, M. Z. 1972. Spectral asymptotics of nonsmooth elliptic operators. I, II. Trudy Moskov. Mat. Obšč., 27, 3–52; ibid. 28 (1973), 3–34. English translation in Trans. Moscow Math. Soc., 27, (1972), 1–52 (1975); ibid., 28, (1973), 1–32 (1975).Google Scholar
Birman, M. Š., and Solomjak, M. Z. 1977a. Asymptotic properties of the spectrum of differential equations. Pages 5–58, i. (loose errata) of: Mathematical Analysis, Vol. 14.Google Scholar
Birman, M. Š., and Solomjak, M. Z. 1977b. Estimates for the singular numbers of integral operators. Uspehi Mat. Nauk, 32(1(193)), 17–84, 271. English translation in Russian Math. Surveys 32, (1977), 15–89.Google Scholar
Birman, M. Š., and Solomjak, M. Z. 1980. Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory. Translated from the Russian by Cezus, F. A.. Amer. Math. Soc. Transl., Series 2, vol. 114. Amer. Math. Soc., Providence, RI.Google Scholar
Birman, M. Sh., and Solomjak, M. Z. 1987. Spectral Theory of Selfadjoint Operators in Hilbert Space. Translated from the 1980 Russian original by Khrushchëv, S. and Peller, V.. Mathematics and its Applications (Soviet Series). Dordrecht: D. Reidel Publishing Co.Google Scholar
Birman, M. Sh., and Solomyak, M. Z. 1991. Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations. Pages 1–55 of: Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989–90). Birman, M. Sh. (editor). Adv. Soviet Math., vol. 7. Amer. Math. Soc., Providence, RI.Google Scholar
Birman, M. Sh., and Solomyak, M. Z. 1992. Schrödinger operator. Estimates for number of bound states as function-theoretical problem. Pages 1–54 of: Spectral Theory of Operators (Novgorod, 1989). Gindikin, S. G. (editor). Amer. Math. Soc. Transl. Ser. 2, vol. 150. Amer. Math. Soc., Providence, RI.Google Scholar
Birman, M. Š., Koplienko, L. S., and Solomjak, M. Z. 1975. Estimates of the spectrum of a difference of fractional powers of selfadjoint operators. Izv. Vysš. Učebn. Zaved. Matematika, 3–10. English translation in Soviet Math. (Iz. VUZ), 19(3), (1975), 1–6.Google Scholar
Birman, M. Sh., Karadzhov, G. E., and Solomyak, M. Z. 1991. Boundedness conditions and spectrum estimates for the operators b(X )a(D) and their analogs. Pages 85–106 of: Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989–90). Birman, M. Sh. (editor). Adv. Soviet Math., vol. 7. Amer. Math. Soc., Providence, RI.Google Scholar
Birman, M. Sh., Laptev, A., and Solomyak, M. 1997. The negative discrete spectrum of the operator (-Δ)l - αV in L2(Rd ) for d even and 2l ≥ d. Ark. Mat., 35(1), 87–126.CrossRefGoogle Scholar
Blanchard, Ph., and Stubbe, J. 1996. Bound states for Schrödinger Hamiltonians: phase space methods and applications. Rev. Math. Phys., 8(4), 503–547.Google Scholar
Blanchard, Ph., Stubbe, J., and Rezende, J. 1987. New estimates on the number of bound states of Schrödinger operators. Lett. Math. Phys., 14(3), 215–225.CrossRefGoogle Scholar
Blankenbecler, R., Goldberger, M.L., and Simon, B. 1977. The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians. Annals of Physics, 108(1), 69–78.CrossRefGoogle Scholar
Bliss, G. A. 1930. An integral inequality. J. Lond. Math. Soc., 5(1), 40–46.CrossRefGoogle Scholar
Bögli, S. 2017. Schrödinger operator with non-zero accumulation points of complex eigenvalues. Comm. Math. Phys., 352(2), 629–639.Google Scholar
Bögli, S., and Cuenin, J-C. 2021. Counterexample to the Laptev–Safronov conjecture. Preprint, available at ArXiv:2109.06135.Google Scholar
Bögli, S„ and Štampach, F. 2020. On Lieb–Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schrödinger operators. J. Spec. Theory, to appear. Preprint, available at ArXiv:2004.09794.Google Scholar
Bohr, N. 1913. The spectra of helium and hydrogen. Nature, 92, 231–232.CrossRefGoogle Scholar
Borichev, A., Golinskii, L., and Kupin, S. 2009. A Blaschke-type condition and its application to complex Jacobi matrices. Bull. Lond. Math. Soc., 41(1), 117–123.Google Scholar
Borichev, A., Frank, R., and Volberg, A. 2022. Counting eigenvalues of Schrödinger operator with complex fast decreasing potential. Adv. Math., 397, 108115.Google Scholar
Born, M., and Jordan, P. 1925. Zur Quantenmechanik. Z. Phys., 34, 858–888.Google Scholar
Brasco, L., De Philippis, G., and Velichkov, B. 2015. Faber–Krahn inequalities in sharp quantitative form. Duke Math. J., 164(9), 1777–1831.Google Scholar
Breuer, J., Simon, B., and Zeitouni, O. 2018. Large deviations and sum rules for spectral theory: a pedagogical approach. J. Spectr. Theory, 8(4), 1551–1581.Google Scholar
Brezis, H. 2011. Functional Analysis, Sobolev Spaces and Partial Differential equations. Universitext. Springer, New York.Google Scholar
Brezis, H., and Lieb, E. H. 1985. Sobolev inequalities with remainder terms. J. Funct. Anal., 62(1), 73–86.Google Scholar
Brossard, J., and Carmona, R. 1986. Can one hear the dimension of a fractal? Comm. Math. Phys., 104(1), 103–122.Google Scholar
Brownell, F. H., and Clark, C. W. 1961. Asymptotic distribution of the eigenvalues of the lower part of the Schrödinger operator spectrum. J. Math. Mech., 10(31–70; addendum), 525–527.Google Scholar
Bucur, D., and Henrot, A. 2019. Maximization of the second non-trivial Neumann eigenvalue. Acta Math., 222(2), 337–361.Google Scholar
Bugliaro, L., Fefferman, C., Fröhlich, J., Graf, G. M., and Stubbe, J. 1997. A Lieb–Thirring bound for a magnetic Pauli Hamiltonian. Comm. Math. Phys., 187(3), 567–582.Google Scholar
Burchard, A., and Thomas, L. E. 2005. On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop. J. Geom. Anal., 15(4), 543–563.Google Scholar
Buslaev, V. S. 1962. Trace formulas for the Schrödinger operator in a three-dimensional space. Dokl. Akad. Nauk SSSR, 143, 1067–1070.Google Scholar
Buslaev, V. S. 1966. The trace formulae and certain asymptotic estimates of the kernel of the resolvent for the Schrödinger operator in three-dimensional space. Pages 82–101 of: Spectral Theory and Wave Processes. Problems of Mathematical Physics, No. 1. Izdat. Leningrad. Univ., Leningrad. English translation in Spectral Theory and Wave Processes. M. Sh. Birman (editor). Topics in Mathematical Physics, vol 1. Springer, Boston, MA.Google Scholar
Buslaev, V. S., and Faddeev, L. D. 1960. Formulas for traces for a singular Sturm–Liouville differential operator. Dokl. Akad. Nauk SSSR, 132, 13–16. English translation in Soviet Math. Dokl., 1, (1960), 451–454.Google Scholar
Buttazzo, G., Guarino Lo Bianco, S., and Marini, M. 2018. Sharp estimates for the anisotropic torsional rigidity and the principal frequency. J. Math. Anal. Appl., 457(2), 1153–1172.CrossRefGoogle Scholar
Calderón, A.-P. 1961. Lebesgue spaces of differentiable functions and distributions. Pages 33–49 of: Proc. Sympos. Pure Math., Vol. IV. Amer. Math. Soc., Providence, RI.Google Scholar
Calogero, F. 1965. Upper and lower limits for the number of bound states in a given central potential. Comm. Math. Phys., 1, 80–88.Google Scholar
Carleman, T. 1935. Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes. Pages 34–44 of: 8. Skand. Mat.-Kongr..Google Scholar
Carleman, T. 1936. Über die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl., 88, 119–132.Google Scholar
Carlen, E. A. 2010. Trace inequalities and quantum entropy: an introductory course. Pages 73–140 of: Entropy and the Quantum. Contemp. Math., vol. 529. Amer. Math. Soc., Providence, RI.Google Scholar
Carlen, E. A., Carrillo, J. A., and Loss, M. 2010. Hardy–Littlewood–Sobolev inequalities via fast diffusion flows. Proc. Natl. Acad. Sci. USA, 107(46), 19696–19701.Google Scholar
Carlen, E. A., Frank, R. L., and Lieb, E. H. 2014. Stability estimates for the lowest eigenvalue of a Schrödinger operator. Geom. Funct. Anal., 24(1), 63–84.Google Scholar
Case, K. M. 1974. Orthogonal polynomials from the viewpoint of scattering theory. J. Mathematical Phys., 15, 2166–2174.CrossRefGoogle Scholar
Case, K. M. 1975. Orthogonal polynomials. II. J. Mathematical Phys., 16, 1435–1440.Google Scholar
Chadan, K. M. 1968. The asymptotic behaviour of the number of bound states of a given potential in the limit of large coupling. Nuovo Cimento, 58 A(1), 191 ff.Google Scholar
Chadan, K. M., Khuri, N. N., Martin, A., and Wu, T. T. 2003. Bound states in one and two spatial dimensions. J. Math. Phys., 44(2), 406–422.Google Scholar
Chang, S.-Y. A., Wilson, J. M., and Wolff, T. H. 1985. Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv., 60(2), 217–246.Google Scholar
Chemin, J.-Y., and Xu, C.-J. 1997. Inclusions de Sobolev en calcul de Weyl–Hörmander et champs de vecteurs sous-elliptiques. Ann. Sci. École Norm. Sup. (4), 30(6), 719–751.Google Scholar
Chepyzhov, V. V., and Ilyin, A. A. 2004. On the fractal dimension of invariant sets: applications to Navier–Stokes equations. Discrete and Continuous Dynamical Systems. Series A, 10(1–2), 117–135. .Google Scholar
Ciesielski, Z. 1970. On the spectrum of the Laplace operator. Comment. Math. Prace Mat., 14, 41–50.Google Scholar
Clark, C. 1967. The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems. SIAM Rev., 9, 627–646.Google Scholar
Coffman, C. V. 1972. Uniqueness of the ground state solution for Δu - u + u3 = 0 and a variational characterization of other solutions. Arch. Rational Mech. Anal., 46, 81–95.CrossRefGoogle Scholar
Colin de Verdière, Y. 1977. Nombre de points entiers dans une famille homothétique de domains de Rn. Ann. Sci. École Norm. Sup. (4), 10(4), 559–575.Google Scholar
Colin de Verdière, Y. 1981. Une formule de traces pour l’opérateur de Schrödinger dans R3. Ann. Sci. École Norm. Sup. (4), 14(1), 27–39.Google Scholar
Colin de Verdière, Y. 1987. Construction de Laplaciens dont une partie finie du spectre est donnée. Ann. Sci. École Norm. Sup. (4), 20(4), 599–615.CrossRefGoogle Scholar
Colin de Verdière, Y. 2010–2011. On the remainder inthe Weyl formula for the Euclidean disk. Séminaire de Théorie Spectrale et Géométrie, 29, 1–13.Google Scholar
Conlon, J. G. 1985. A new proof of the Cwikel–Lieb–Rosenbljum bound. Rocky Mountain J. Math., 15(1), 117–122.Google Scholar
Constantin, P., Foias, C., and Temam, R. 1985. Attractors representing turbulent flows. Mem. Amer. Math. Soc., 53(314).Google Scholar
Cordero-Erausquin, D., Nazaret, B., and Villani, C. 2004. A mass-transportation ap- proach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math., 182(2), 307–332.CrossRefGoogle Scholar
Cornfeld, I. P., Fomin, S. V., and Sinaĭ, Ya. G. 1982. Ergodic Theory. Translated from the Russian by Sosinskiĭ, A. B.. Grundlehren der Mathematischen Wissenschaften, vol. 245. Springer-Verlag, New York.Google Scholar
Courant, R. 1920. Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik. Math. Z., 7(1–4), 1–57.Google Scholar
Courant, R. 1922. Über die Lösungen der Differentialgleichungen der Physik. Math. Ann., 85(1), 280–325.Google Scholar
Courant, R. 1925. Über direkte Methoden bei Variations- und Randwertproblemen. Jahresber. Dtsch. Math.-Ver., 34, 90–117.Google Scholar
Courant, R., and Hilbert, D. 1953. Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc., New York.Google Scholar
Courant, R., and Hilbert, D. 1962. Methods of Mathematical Physics. Vol. II. Partial Differential Equations. Reprinted 1989 in paperback in Wiley Classics Library. John Wiley & Sons, Inc., New York.Google Scholar
Crum, M. M. 1955. Associated Sturm–Liouville systems. Q. J. Math., Oxf. II. Ser., 6, 121–125.Google Scholar
Cuenin, J.-C. 2020. Improved eigenvalue bounds for Schrödinger operators with slowly decaying potentials. Comm. Math. Phys., 376(3), 2147–2160.Google Scholar
Cuenin, J.-C., Laptev, A., and Tretter, C. 2014. Eigenvalue estimates for non-selfadjoint Dirac operators on the real line. Ann. Henri Poincaré, 15(4), 707–736.Google Scholar
Cwikel, M. 1977. Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. (2), 106(1), 93–100.CrossRefGoogle Scholar
Cycon, H. L., Froese, R. G., Kirsch, W., and Simon, B. 1987. Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics. Springer-Verlag, Berlin.Google Scholar
Damanik, D., and Remling, C. 2007. Schrödinger operators with many bound states. Duke Math. J., 136(1), 51–80.Google Scholar
Darboux, G. 1882. Sur une proposition relative aux équations linéaires. C. R. Acad. Sci., Paris, 94, 1456–1459.Google Scholar
Daubechies, I. 1983. An uncertainty principle for fermions with generalized kinetic energy. Comm. Math. Phys., 90(4), 511–520.Google Scholar
Davies, E. B. 1995. Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics, vol. 42. Cambridge University Press, Cambridge.Google Scholar
Davies, E. B., and Nath, J. 2002. Schrödinger operators with slowly decaying potentials. J. Computat. Appl. Math.. 148(1), 1–28.Google Scholar
Davies, E. B., and Simon, B. 1992. Spectral properties of Neumann Laplacian of horns. Geom. Funct. Anal., 2(1), 105–117.Google Scholar
de Guzmán, M. 1975. Differentiation of Integrals in Rn. Lecture Notes in Mathematics, Vol. 481. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón. Springer-Verlag, Berlin and New York.CrossRefGoogle Scholar
de la Bretèche, R. 1999. Preuve de la conjecture de Lieb-Thirring dans le cas des potentiels quadratiques strictement convexes. Ann. Inst. H. Poincaré Phys. Théor., 70(4), 369–380.Google Scholar
de Wet, J. S., and Mandl, F. 1950. On the asymptotic distribution of eigenvalues. Proc. Roy. Soc. London Ser. A, 200, 572–580.Google Scholar
Deift, P., and Killip, R. 1999. On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Comm. Math. Phys., 203(2), 341–347.Google Scholar
Deift, P., and Trubowitz, E. 1979. Inverse scattering on the line. Comm. Pure Appl. Math., 32(2), 121–251.CrossRefGoogle Scholar
Deift, P. A. 1978. Applications of a commutation formula. Duke Math. J., 45(2), 267–310.Google Scholar
Demirel, S. 2012. Spectral Theory of Quantum Graphs. Dissertation. University of Stuttgart.Google Scholar
Demuth, M., Hansmann, M., and Katriel, G. 2013. Eigenvalues of non-selfadjoint operators: a comparison of two approaches. Pages 107–163 of: Mathematical Physics, Spectral Theory and Stochastic Analysis. Demuth, M. and Kirsch, W. (editors). Oper. Theory Adv. Appl., vol. 232. Birkhäuser/Springer Basel AG, Basel.CrossRefGoogle Scholar
Deny, J., and Lions, J. L. 1954. Les espaces du type de Beppo Levi. Ann. Inst. Fourier (Grenoble), 5, 305–370.Google Scholar
Denzler, J. 2015. Existence and regularity for a curvature dependent variational problem. Trans. Amer. Math. Soc., 367(6), 3829–3845.Google Scholar
Dimassi, M., and Sjöstrand, J. 1999. Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series, vol. 268. Cambridge University Press, Cambridge.Google Scholar
Dolbeault, J., Felmer, P., Loss, M., and Paturel, E. 2006. Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems. J. Funct. Anal., 238(1), 193–220.Google Scholar
Dolbeault, J., Laptev, A., and Loss, M. 2008. Lieb–Thirring inequalities with improved constants. J. Eur. Math. Soc. (JEMS), 10(4), 1121–1126.Google Scholar
Duistermaat, J. J., and Guillemin, V. W. 1975. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math., 29(1), 39–79.Google Scholar
Dyson, F. J., and Lenard, A. 1967. Stability of matter. I. J. Math. Phys., 8(3), 423–434.Google Scholar
Eckart, C. 1930. The penetration of a potential barrier by electrons. Phys. Rev., II. Ser., 35, 1303–1309.Google Scholar
Eden, A., and Foias, C. 1991. A simple proof of the generalized Lieb–Thirring inequalities in one-space dimension. J. Math. Anal. Appl., 162(1), 250–254.Google Scholar
Edmunds, D. E., and Evans, W. D. 2018. Elliptic Differential Operators and Spectral Analysis. Springer Monographs in Mathematics. Springer, Cham.CrossRefGoogle Scholar
Egorov, Yu. V., and Kondrat’ev, V. A. 1992. Estimates of the negative spectrum of an elliptic operator. Pages 111–140 of: Spectral Theory of Operators (Novgorod, 1989). Gindikin, S. G. (editor). Amer. Math. Soc. Transl. Ser. 2, vol. 150. Amer. Math. Soc., Providence, RI.Google Scholar
Egorov, Yu. V., and Kondratiev, V. A. 1995. On moments of negative eigenvalues of an elliptic operator. Math. Nachr., 174, 73–79.Google Scholar
Egorov, Yu., and Kondratiev, V. 1996. On Spectral Theory of Elliptic Operators. Oper. Theory Adv. Appl., vol. 89. Birkhäuser Verlag, Basel.Google Scholar
Ekholm, T., and Frank, R. L. 2006. On Lieb–Thirring inequalities for Schrödinger operators with virtual level. Comm. Math. Phys., 264(3), 725–740.Google Scholar
Ekholm, T., and Frank, R. L. 2008. Lieb–Thirring inequalities on the half-line with critical exponent. J. Eur. Math. Soc. (JEMS), 10(3), 739–755.Google Scholar
El Soufi, A., and Ilias, S. 1986. Immersions minimales, première valeur propre du Laplacien et volume conforme. Math. Ann., 275(2), 257–267.Google Scholar
Epstein, P. S. 1930. Reflection of waves in an inhomogeneous absorbing medium. Proc. Natl. Acad. Sci. USA, 16, 627–637.Google Scholar
Erdős, L. 1995. Magnetic Lieb–Thirring inequalities. Comm. Math. Phys., 170(3), 629–668.Google Scholar
Erdős, L., and Solovej, J. P. 1997. Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. II. Leading order asymptotic estimates. Comm. Math. Phys., 188(3), 599–656.Google Scholar
Erdős, L., and Solovej, J.P. 1999. Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields. I. Nonasymptotic Lieb–Thirring-type estimate. Duke Math. J., 96(1), 127–173.CrossRefGoogle Scholar
Erdős, L., Loss, M., and Vougalter, V. 2000. Diamagnetic behavior of sums of Dirichlet eigenvalues. Ann. Inst. Fourier (Grenoble), 50(3), 891–907.Google Scholar
Evans, L. C. 2010. Partial Differential Equations. Second edn. Graduate Studies in Mathematics, vol. 19. Amer. Math. Soc., Providence, RI.Google Scholar
Exner, P., and Šeba, P. 1989. Bound states in curved quantum waveguides. J. Math. Phys., 30(11), 2574–2580.Google Scholar
Exner, P., Linde, H., and Weidl, T. 2004. Lieb-Thirring inequalities for geometrically induced bound states. Lett. Math. Phys., 70(1), 83–95.Google Scholar
Exner, P., Laptev, A., and Usman, M. 2014. On some sharp spectral inequalities for Schrödinger operators on semiaxis. Comm. Math. Phys., 326(2), 531–541.Google Scholar
Faber, G. 1923. Beweis, daß unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Münch. Ber., 169–172.Google Scholar
Faddeev, L. D. 1957. An expression for the trace of the difference between two singular differential operators of the Sturm–Liouville type. Dokl. Akad. Nauk SSSR (N.S.), 115, 878–881.Google Scholar
Faddeev, L. D. 1958. On the relation between S -matrix and potential for the onedimensional Schrödinger operator. Dokl. Akad. Nauk SSSR (N.S.), 121, 63–66.Google Scholar
Faddeev, L. D. 1964. Properties of the S-matrix of the one-dimensional Schrödinger equation. Trudy Mat. Inst. Steklov., 73, 314–336. English translation in: Amer. Math. Soc. Transl. (Ser. 2), 65, (1967), 139–166.Google Scholar
Fan, K. 1949. On a theorem of Weyl concerning eigenvalues of linear transformations. I. Proc. Nat. Acad. Sci. USA, 35, 652–655.Google Scholar
Farmer, J. D., Ott, E., and Yorke, J. A. 1983. The dimension of chaotic attractors. Phys. D, 7(1–3), 153–180.Google Scholar
Federer, H., and Fleming, W. H. 1960. Normal and integral currents. Ann. of Math. (2), 72, 458–520.Google Scholar
Fefferman, C. L. 1983. The uncertainty principle. Bull. Amer. Math. Soc. (N.S.), 9(2), 129–206.Google Scholar
Filonov, N. 2004. On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator. Algebra i Analiz, 16(2), 172–176. English translation in St. Petersburg Math. J., 16(2), 413–416, (2005).Google Scholar
Filonov, N. 2022. On the Pólya conjecture for circular sectors and for balls. Preprint, available at ArXiv:2208.03463.Google Scholar
Fischer, E. 1905. Über quadratische Formen mit reellen Koeffizienten. Monatsh. Math. Phys., 16(1), 234–249.Google Scholar
Fleckinger-Pellé, J., and Vassiliev, D. G. 1993. An example of a two-term asymptotics for the “counting function” of a fractal drum. Trans. Amer. Math. Soc., 337(1), 99–116.Google Scholar
Flügge, S. 1999. Practical Quantum Mechanics. Springer-Verlag, Berlin and Heidelberg.Google Scholar
Folland, G. B. 1999. Real Analysis. Modern Techniques and their Applications. Second edn. John Wiley & Sons, Inc., New York.Google Scholar
Förster, C., and Weidl, T. 2011. Trapped modes in an elastic plate with a hole. Algebra i Analiz , 23 (1), 255–288. English translation in St. Petersburg Math. J. , 3 (1), 179–202, (2012).Google Scholar
Förster, C. 2008. Trapped modes for an elastic plate with a perturbation of Young’s modulus. Comm. Partial Differential Equations, 33(7–9), 1339–1367.CrossRefGoogle Scholar
Förster, C., and Östensson, J. 2008. Lieb–Thirring inequalities for higher order differential operators. Math. Nachr., 281(2), 199–213.Google Scholar
Frank, R. L. 2009a. Remarks on eigenvalue estimates and semigroup domination. Pages 63–86 of: Spectral and Scattering Theory for Quantum Magnetic Systems. Briet, Philippe, Germinet, François, and Raikov, Georgi (editors). Contemp. Math., vol. 500. Amer. Math. Soc., Providence, RI.Google Scholar
Frank, R. L. 2009b. A simple proof of Hardy–Lieb–Thirring inequalities. Comm. Math. Phys., 290(2), 789–800.Google Scholar
Frank, R. L. 2011. Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc., 43(4), 745–750.CrossRefGoogle Scholar
Frank, R. L. 2014. Cwikel’s theorem and the CLR inequality. J. Spectr. Theory, 4(1), 1–21.Google Scholar
Frank, R. L. 2018a. Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Amer. Math. Soc., 370(1), 219–240.Google Scholar
Frank, R. L. 2018b. Eigenvalue bounds for the fractional Laplacian: a review. Pages 210–235 of: Recent Developments in Nonlocal Theory. Palatucci, G. and Kuusi, T. (editors). De Gruyter, Berlin.Google Scholar
Frank, Rupert L. 2021. The Lieb–Thirring inequalities: Recent results and open problems. Pages 45–86 of: Nine Mathematical Challenges – An Elucidation. Proc. Sympos. Pure Math., vol. 104. Kechris, A., Makarov, N., Ramakrishnan, D., and Zhu, X. (editors). Amer. Math. Soc., Providence, RI.Google Scholar
Frank, R. L. 2022. Weyl’s law under minimal assumptions. Preprint, available at ArXiv:2202.00323.Google Scholar
Frank, R. L., and Geisinger, L. 2011. Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain. Pages 138–147 of: Mathematical Results in Quantum Physics. Exner, P. (editor). World Sci. Publ., Hackensack, NJ.Google Scholar
Frank, R. L., and Geisinger, L. 2016. Refined semiclassical asymptotics for fractional powers of the Laplace operator. J. Reine Angew. Math., 712, 1–37.Google Scholar
Frank, R. L., and Laptev, A. 2008. Spectral inequalities for Schrödinger operators with surface potentials. Pages 91–102 of: Spectral Theory of Differential Operators. Suslina, T. and Yafaev, D. (editors). Amer. Math. Soc. Transl. Ser. 2, vol. 225. Amer. Math. Soc., Providence, RI.Google Scholar
Frank, R. L., and Laptev, A. 2010. Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group. Int. Math. Res. Not., 2010(15), 2889–2902.Google Scholar
Frank, R. L., and Larson, S. 2020. Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain. J. Reine Angew. Math., 766, 195–228.Google Scholar
Frank, R. L., and Lieb, E. H. 2010. Inversion positivity and the sharp Hardy-Littlewood- Sobolev inequality. Calc. Var. Partial Differential Equations, 39(1–2), 85–99.Google Scholar
Frank, R. L., and Lieb, E. H. 2012. A new, rearrangement-free proof of the sharp Hardy–Littlewood–Sobolev inequality. Pages 55–67 of: Spectral Theory, Function Spaces and Inequalities. Brown, B. M., Lang, J., and Wood, I. G. (editors). Oper. Theory Adv. Appl., vol. 219. Birkhäuser/Springer Basel AG, Basel.Google Scholar
Frank, R. L., and Pushnitski, A. 2015. Trace class conditions for functions of Schrödinger operators. Comm. Math. Phys., 335(1), 477–496.CrossRefGoogle Scholar
Frank, R. L., and Sabin, J. 2017a. Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Amer. J. Math., 139(6), 1649–1691.Google Scholar
Frank, R. L., and Sabin, J. 2017b. Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces. Adv. Math., 317, 157–192.Google Scholar
Frank, R. L., and Sabin, J. 2017c. The Stein–Tomas inequality in trace ideals. Pages Exp. No. XV, 12 of: Séminaire Laurent Schwartz – Équations aux Dérivées Partielles et Applications. Année 2015–2016. Ed. Éc. Polytech., Palaiseau.Google Scholar
Frank, R. L., and Sabin, J. 2022. Sharp Weyl laws with singular potentials. Pure and Appl. Anal., to appear. Preprint, available at ArXiv:2007.04284.Google Scholar
Frank, R. L., and Seiringer, R. 2008. Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal., 255(12), 3407–3430.Google Scholar
Frank, R. L., and Seiringer, R. 2012. Lieb–Thirring inequality for a model of particles with point interactions. J. Math. Phys., 53(9), 095201, 11.Google Scholar
Frank, R. L., and Simon, B. 2011. Critical Lieb–Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices. Duke Math. J., 157(3), 461–493.Google Scholar
Frank, R. L., and Simon, B. 2017. Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory, 7(3), 633–658.Google Scholar
Frank, R. L., Laptev, A., Lieb, E. H., and Seiringer, R. 2006. Lieb–Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys., 77(3), 309–316.Google Scholar
Frank, R. L., Lieb, E. H., and Seiringer, R. 2007. Number of bound states of Schrödinger operators with matrix-valued potentials. Lett. Math. Phys., 82(2–3), 107–116.Google Scholar
Frank, R. L., Simon, B., and Weidl, T. 2008a. Eigenvalue bounds for perturbations of Schrödinger operators and Jacobi matrices with regular ground states. Comm. Math. Phys., 282(1), 199–208.CrossRefGoogle Scholar
Frank, R. L., Lieb, E. H., and Seiringer, R. 2008b. Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc., 21(4), 925–950.Google Scholar
Frank, R. L., Loss, M., and Weidl, T. 2009. Pólya’s conjecture in the presence of a constant magnetic field. J. Eur. Math. Soc. (JEMS), 11(6), 1365–1383.Google Scholar
Frank, R. L., Lieb, E. H., and Seiringer, R. 2010. Equivalence of Sobolev inequalities and Lieb–Thirring inequalities. Pages 523–535 of: XVIth International Congress on Mathematical Physics. Exner, P. (editor). World Sci. Publ., Hackensack, NJ.Google Scholar
Frank, R. L., Lewin, M., Lieb, E. H., and Seiringer, R. 2013. A positive density analogue of the Lieb–Thirring inequality. Duke Math. J., 162(3), 435–495.CrossRefGoogle Scholar
Frank, R. L., Lewin, M., Lieb, E. H., and Seiringer, R. 2014. Strichartz inequality for orthonormal functions. J. Eur. Math. Soc. (JEMS), 16(7), 1507–1526.Google Scholar
Frank, R. L., Laptev, A., and Safronov, O. 2016. On the number of eigenvalues of Schrödinger operators with complex potentials. J. Lond. Math. Soc. (2), 94(2), 377–390.CrossRefGoogle Scholar
Frank, R. L., Gontier, D., and Lewin, M. 2021a. The periodic Lieb–Thirring inequality. Pages 135–154 of: Partial Differential Equations, Spectral Theory, and Mathematical Physics: The Ari Laptev Anniversary Volume. Exner, P., Frank, R. L., Gesztesy, F., Holden, H., Weidl, T. (editors). European Mathematical Society Publishing House, Zürich.Google Scholar
Frank, R. L., Gontier, D., and Lewin, M. 2021b. The nonlinear Schrödinger equation for orthonormal functions II: Application to Lieb–Thirring Inequalities. Comm. Math. Phys., 384(3), 1783–1828.Google Scholar
Frank, R. L., Gontier, D., and Lewin, M. 2021c. Optimizers for the finite-rank Lieb–Thirring inequality. Preprint, available at ArXiv:2109.05984.Google Scholar
Frank, R. L., Hundertmark, D., Jex, M., and Nam, P.T. 2021d. The Lieb–Thirring inequality revisited. J. Eur. Math. Soc. (JEMS), 23(8), 2583–2600.Google Scholar
Frank, R. L., Laptev, A., and Weidl, T. 2022. An improved one-dimensional Hardy inequality. J. Math. Sciences, to appear. Preprint, available at ArXiv:2204.00877.Google Scholar
Friedlander, L. 1991. Some inequalities between Dirichlet and Neumann eigenvalues. Arch. Rational Mech. Anal., 116(2), 153–160.Google Scholar
Friedman, A. 1970. Foundations of Modern Analysis. Reprinted 1982 in paperback by Dover Publications, Inc., New York.Google Scholar
Friedrichs, K. 1928. Die Randwert-und Eigenwertprobleme aus der Theorie der elastischen Platten. (Anwendung der direkten Methoden der Variationsrechnung). Math. Ann., 98(1), 205–247.Google Scholar
Friedrichs, K. 1934a. Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren. Math. Ann., 109(1), 465–487.Google Scholar
Friedrichs, K. 1934b. Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren. Math. Ann., 109(1), 685–713.CrossRefGoogle Scholar
Gagliardo, E. 1958. Proprietà di alcune classi di funzioni in più variabili. Ric. Mat., 7, 102–137.Google Scholar
Gagliardo, E. 1959. Ulteriori proprieta di alcune classi di funzioni in piu variabili. Ric. Mat., 8, 24–51.Google Scholar
Gamboa, F., Nagel, J., and Rouault, A. 2016. Sum rules via large deviations. J. Funct. Anal., 270(2), 509–559.Google Scholar
Gårding, L. 1951. The asymptotic distribution of the eigenvalues and eigenfunctions of a general vibration problem. Kungl. Fysiografiska Sällskapets i Lund Förhandlingar [Proc. Roy. Physiog. Soc. Lund], 21(11), 10.Google Scholar
Gårding, L. 1953. Dirichlet’s problem for linear elliptic partial differential equations. Math. Scand., 1, 55–72.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M. 1967. Method for solving the Korteweg–deVries equation. Phys. Rev. Lett., 19(Nov), 1095–1097.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M. 1974. Korteweg–deVries equation and generalization. VI. Methods for exact solution. Comm. Pure Appl. Math., 27, 97–133.Google Scholar
Garnett, J. B. 2007. Bounded Analytic Functions. Revised first edn. Graduate Texts in Mathematics, vol. 236. Springer, New York.Google Scholar
Geisinger, L., and Weidl, T. 2011. Sharp spectral estimates in domains of infinite volume. Rev. Math. Phys., 23(6), 615–641.Google Scholar
Geisinger, L., Laptev, A., and Weidl, T. 2011. Geometrical versions of improved Berezin–Li–Yau inequalities. J. Spectr. Theory, 1(1), 87–109.Google Scholar
Gel’fand, I. M. 1956. On identities for eigenvalues of a differential operator of second order. Uspehi Mat. Nauk (N.S.), 11(1(67)), 191–198.Google Scholar
Gel’fand, I. M., and Levitan, B. M. 1953. On a simple identity for the characteristic values of a differential operator of the second order. Doklady Akad. Nauk SSSR (N.S.), 88, 593–596.Google Scholar
Gesztesy, F. 1993. A complete spectral characterization of the double commutation method. J. Funct. Anal., 117(2), 401–446.Google Scholar
Gesztesy, F., and Teschl, G. 1996. On the double commutation method. Proc. Amer. Math. Soc., 124(6), 1831–1840.Google Scholar
Gesztesy, F., Simon, B., and Teschl, G. 1996. Spectral deformations of one-dimensional Schrödinger operators. J. Anal. Math., 70, 267–324.Google Scholar
Gidas, B., Ni, W. M., and Nirenberg, L. 1981. Symmetry of positive solutions of nonlinear elliptic equations in Rn . Pages 369–402 of: Mathematical Analysis and Applications, Part A. Nachbin, L. (editor). Adv. in Math. Suppl. Stud., vol. 7. Academic Press, New York and London.Google Scholar
Gilbarg, D., and Trudinger, N. S. 1998. Elliptic Partial Differential Equations of Second Order. Paperback reprint 1998. Classics in Mathematics. Springer-Verlag, Berlin.Google Scholar
Glaser, V., Martin, A., Grosse, H., and Thirring, W. 1976. A family of optimal conditions for the absence of bound states in a potential. Pages 169–194 of: Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann. Lieb, E. H., Simon, B., and Wightman, A. S. (editors). Princeton Series in Physics vol. 58. Princeton University Press, Princeton, NJ.Google Scholar
Glaser, V., Grosse, H., and Martin, A. 1978. Bounds on the number of eigenvalues of the Schrödinger operator. Comm. Math. Phys., 59(2), 197–212.Google Scholar
Glazman, I. M. 1966. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators. Translated from the Russian by the IPST staff. Israel Program for Scientific Translations, Jerusalem, 1965; Daniel Davey & Co., Inc., New York.Google Scholar
Gordon, C., Webb, D., and Wolpert, S. 1992a. Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math., 110(1), 1–22.CrossRefGoogle Scholar
Gordon, C., Webb, D. L., and Wolpert, S. 1992b. One cannot hear the shape of a drum. Bull. Amer. Math. Soc. (N.S.), 27(1), 134–138.Google Scholar
Gontier, D., Lewin, M., and Nazar, F. Q. 2021. The nonlinear Schrödinger equation for orthonormal functions: existence of ground states. Arch. Rational Mech. Anal., 240(3), 1203–1254.CrossRefGoogle Scholar
Grebenkov, D. S., and Nguyen, B.-T. 2013. Geometrical structure of Laplacian eigenunctions. SIAM Rev., 55(4), 601–667.Google Scholar
Grigor’yan, A., and Nadirashvili, N. 2015. Negative eigenvalues of two-dimensional Schrödinger operators. Arch. Rational Mech. Anal., 217(3), 975–1028.Google Scholar
Grigor’yan, A., Netrusov, Y., and Yau, S.-T. 2004. Eigenvalues of elliptic operators and geometric applications. Pages 147–217 of: Surveys in Differential Geometry. Vol. IX. Grigor’yan, A. and Yau, S.-T. (editors). International Press, Somerville, MA.Google Scholar
Grigor’yan, A., Nadirashvili, N., and Sire, Y. 2016. A lower bound for the number of negative eigenvalues of Schrödinger operators. J. Differential Geom., 102(3), 395–408.Google Scholar
Grisvard, P. 1985. Elliptic Problems in Nonsmooth Domains. Paperback reprint, with a foreword by Susanne C. Brenner 2011. Classics in Applied Mathematics, vol. 69. SIAM, Philadelphia, PA.Google Scholar
Grünbaum, B., and Shephard, G. C. 1987. Tilings and Patterns. W. H. Freeman and Company, New York.Google Scholar
Hänel, A., and Weidl, T. 2017. Spectral asymptotics for the Dirichlet Laplacian with a Neumann window via a Birman–Schwinger analysis of the Dirichlet-to-Neumann operator. Pages 315–352 of: Functional Analysis and Operator Theory for Quantum Physics. Dittrich, J., Kovarik, H.. and Laptev, A. (editors). EMS Ser. Congr. Rep. Eur. Math. Soc., Zürich.Google Scholar
Hainzl, C., Hamza, E., Seiringer, R., and Solovej, J. P. 2008. The BCS functional for general pair interactions. Comm. Math. Phys., 281(2), 349–367.Google Scholar
Hajłasz, P., Koskela, P., and Tuominen, H. 2008. Sobolev embeddings, extensions and measure density condition. J. Funct. Anal., 254(5), 1217–1234.Google Scholar
Hansmann, M. 2011. An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators. Lett. Math. Phys., 98(1), 79–95.CrossRefGoogle Scholar
Hansson, A. M., and Laptev, A. 2008. Sharp spectral inequalities for the Heisenberg Laplacian. Pages 100–115 of: Groups and Analysis. Tent, K. (editor). London Math. Soc. Lecture Note Ser., vol. 354. Cambridge University Press, Cambridge.Google Scholar
Hardy, G. H. 1919. Notes on some points in the integral calculus. LI. (On Hilbert’s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals.). Messenger of Mathematics, 48, 107–112.Google Scholar
Hardy, G. H. 1925. Notes on some points in the integral calculus. LX. Messenger of Mathematics, 54, 150–156.Google Scholar
Hardy, G. H. 1928. Note on some points in the integral calculus LXIV. Messenger of Mathematics, 57, 12–16.Google Scholar
Hardy, G. H., Littlewood, J. E., and Pólya, G. 1942. Inequalities. Reprinted in paperback in Cambridge Mathematical Library (1988). Cambridge University Press, Cambridge.Google Scholar
Harrell, E. M. II, and Stubbe, J. 1997. On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Amer. Math. Soc., 349(5), 1797–1809.CrossRefGoogle Scholar
Harrell, E. M. II, and Stubbe, J. 2010. Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators. SIAM J. Math. Anal., 42(5), 2261–2274.Google Scholar
Harrell, E. M. II, and Stubbe, J. 2011. Trace identities for commutators, with applications to the distribution of eigenvalues. Trans. Amer. Math. Soc., 363(12), 6385–6405.Google Scholar
Harrell, E. M. II, and Stubbe, J. 2018. Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian. J. Spectr. Theory, 8(4), 1529–1550.CrossRefGoogle Scholar
Hebey, E. 1996. Sobolev Spaces on Riemannian Manifolds. Lecture Notes in Mathematics, vol. 1635. Springer-Verlag, Berlin.Google Scholar
Heisenberg, W. 1925. Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Phys., 33, 879–893.Google Scholar
Helffer, B., and Robert, D. 1983. Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles. J. Funct. Anal., 53(3), 246–268.Google Scholar
Helffer, B., and Robert, D. 1990a. Riesz means of bound states and semiclassical limit connected with a Lieb–Thirring’s conjecture. Asymptotic Anal., 3(2), 91–103.Google Scholar
Helffer, B., and Robert, D. 1990b. Riesz means of bounded states and semi-classical limit connected with a Lieb–Thirring conjecture. II. Ann. Inst. H. Poincaré Phys. Théor., 53(2), 139–147.Google Scholar
Helffer, B., and Sjöstrand, J. 1990. On diamagnetism and de Haas-van Alphen effect. Ann. Inst. H. Poincaré Phys. Théor., 52(4), 303–375.Google Scholar
Helffer, B.. 2013. Spectral Theory and its Applications. Cambridge Studies in Advanced Mathematics, vol. 139. Cambridge University Press, Cambridge.Google Scholar
Helffer, B., and Persson, Sundqvist M. 2016. On nodal domains in Euclidean balls. Proc. Amer. Math. Soc., 144(11), 4777–4791.Google Scholar
Hempel, R., Seco, L. A., and Simon, B. 1991. The essential spectrum of Neumann Laplacians on some bounded singular domains. J. Funct. Anal., 102(2), 448–483.Google Scholar
Henrot, A. 2006. Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel.Google Scholar
Henrot, A. (editor). 2017. Shape Optimization and Spectral Theory. De Gruyter, Berlin.CrossRefGoogle Scholar
Herbst, I. W., and Zhao, Z. X. 1988. Sobolev spaces, Kac-regularity, and the Feynman–Kac formula. Pages 171–191 of: Seminar on Stochastic Processes, (Princeton, NJ, 1987). Çinlar, E., Chung, K. L., Getoor, R. K., and Glover, J. (editors). Progr. Probab. Statist., vol. 15. Birkhäuser Boston, Boston, MA.Google Scholar
Hersch, J. 1960. Sur la fréquence fondamentale d’une membrane vibrante: évaluations par défaut et principe de maximum. Z. Angew. Math. Phys., 11, 387–413.Google Scholar
Hilbert, D. 1906. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Vierte Mitteilung. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 1906, 157–227.Google Scholar
Hoang, V., Hundertmark, D., Richter, J., and Vugalter, S. 2017. Quantitative bounds versus existence of weakly coupled bound states for Schrödinger type operators. Preprint, available at ArXiv:1610.09891.Google Scholar
Hoffmann-Ostenhof, M., and Hoffmann-Ostenhof, T. 1977. “Schrödinger inequalities” and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A (3), 16(5), 1782–1785.Google Scholar
Hong, I. 1954. On an inequality concerning the eigenvalue problem of membrane. Ko¯ dai Math. Sem. Rep., 6, 113–114.Google Scholar
Hoppe, J., Laptev, A., and Östensson, J. 2006. Solitons and the removal of eigenvalues for fourth-order differential operators. Int. Math. Res. Not., Art. ID 85050, 14.Google Scholar
Hörmander, L. 1968. The spectral function of an elliptic operator. Acta Math., 121, 193–218.Google Scholar
Hörmander, L. 1983. The Analysis of Linear Partial Differential operators. II, Differential Operators with Constant Coefficients. Reprinted 2005 in Classics in Mathematics. Springer-Verlag, Berlin.Google Scholar
Hörmander, L. 1990. The Analysis of Linear Partial Differential operators. I, Distribution Theory and Fourier Analysis. Second edn. Reprinted 2003 in Classics in Mathematics. Springer-Verlag, Berlin.Google Scholar
Hörmander, L. 1994a. The Analysis of Linear Partial Differential operators. III, Pseudodifferential Operators. Reprinted 2007 in Classics in Mathematics. Springer-Verlag, Berlin.Google Scholar
Hörmander, L. 1994b. The Analysis of Linear Partial Differential operators. IV, Fourier Integral Operators. Reprinted 2009 in Classics in Mathematics. Springer-Verlag, Berlin.Google Scholar
Hryniv, R., and Mykytyuk, Y. 2021. On the first trace formula for Schrödinger operators. J. Spect. Theory, 11, 489–507.CrossRefGoogle Scholar
Hryniv, R., Melnyk, B., and Mykytyuk, Y. 2021. Inverse scattering for reflectionless Schrödinger operators with integrable potentials and generalized soliton solutions for the KdV equation. Ann. Henri Poincaré, 22(2), 487–527.Google Scholar
Hundertmark, D. 2002. On the number of bound states for Schrödinger operators with operator-valued potentials. Ark. Mat., 40(1), 73–87.Google Scholar
Hundertmark, D., and Simon, B. 2002. Lieb–Thirring inequalities for Jacobi matrices. J. Approx. Theory, 118(1), 106–130.CrossRefGoogle Scholar
Hundertmark, D., Lieb, E. H., and Thomas, L. E. 1998. A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. Adv. Theor. Math. Phys., 2(4), 719–731.Google Scholar
Hundertmark, D., Laptev, A., and Weidl, T. 2000. New bounds on the Lieb–Thirring constants. Invent. Math., 140(3), 693–704.Google Scholar
Hundertmark, D., Kunstmann, P., Ried, T., and Vugalter, S.. 2018. Cwikel’s bound reloaded. Preprint, available at ArXiv:1809.05069.Google Scholar
Il’in, A. A. 2005. Lieb–Thirring integral inequalities and their applications to attractors of Navier–Stokes equations. Mat. Sb., 196(1), 33–66. English translation in Sb. Math. 196(1–2), 29–61, (2005).Google Scholar
Il’in, A. A. 2009. On the spectrum of the Stokes operator. Funktsional. Anal. i Prilozhen., 43(4), 14–25. English translation in Funct. Anal. Appl. 43(4), 254–263, (2009).Google Scholar
Ilyin, A., Laptev, A., Loss, M., and Zelik, S. 2016. One-dimensional interpolation inequalities, Carlson–Landau inequalities, and magnetic Schrödinger operators. Int. Math. Res. Not., 1190–1222.Google Scholar
Ivriĭ, V. Ja. 1980a. The second term of the spectral asymptotics for a Laplace–Beltrami operator on manifolds with boundary. Funktsional. Anal. i Prilozhen., 14(2), 25–34. English translation in Functional Anal. Appl., 14(2), 98–106, (1980).Google Scholar
Ivriĭ, V. Ja. 1980b. The second term of the spectral asymptotics for the Laplace–Beltrami operator on manifolds with boundary and for elliptic operators acting in vector bundles. Dokl. Akad. Nauk SSSR, 250(6), 1300–1302. English translation in Soviet Math. Dokl., 20(1), 300–302, (1980).Google Scholar
Ivrii, V. 1998. Microlocal Analysis and Precise Spectral Asymptotics. Springer Monographs in Mathematics. Springer-Verlag, Berlin.Google Scholar
Ivrii, V. 2019a. Microlocal Analysis, Sharp Spectral Asymptotics and Applications. I, Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics. Springer, Cham.CrossRefGoogle Scholar
Ivrii, V. 2019b. Microlocal Analysis, Sharp Spectral Asymptotics and Applications. II, Functional Methods and Eigenvalue Asymptotics. Springer, Cham.Google Scholar
Ivrii, V. 2019c. Microlocal Analysis, Sharp Spectral Asymptotics and Applications. III, Magnetic Schrödinger Operator 1. Springer, Cham.Google Scholar
Ivrii, V. 2019d. Microlocal Analysis, Sharp Spectral Asymptotics and Applications. IV, Magnetic Schrödinger operator 2. Springer, Cham.Google Scholar
Jacobi, C. G. J. 1837. Zur Theorie der Variationsrechnung und der Differentialgleichungen. J. Reine Angew. Math., 17, 68–82.Google Scholar
Jakšić, V., Molčanov, S., and Simon, B. 1992. Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps. J. Funct. Anal., 106(1), 59–79.CrossRefGoogle Scholar
Jensen, A. 1980. Spectral properties of Schrödinger operators and time-decay of the wave functions results in L2 (Rm ), m ≥ 5. Duke Math. J., 47(1), 57–80.Google Scholar
Jensen, A. 1984. Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in L2 (R4 ). J. Math. Anal. Appl., 101(2), 397–422.CrossRefGoogle Scholar
Jensen, A., and Kato, T. 1979. Spectral properties of Schrödinger operators and timedecay of the wave functions. Duke Math. J., 46(3), 583–611.Google Scholar
Jones, P. W. 1981. Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math., 147(1–2), 71–88.Google Scholar
Jost, R., and Pais, A. 1951. On the scattering of a particle by a static potential. Phys. Rev. (2), 82, 840–851.Google Scholar
Kac, I. S., and Kreĭn, M. G. 1958. Criteria for the discreteness of the spectrum of a singular string. Izv. Vysš. Učebn. Zaved. Matematika, 1958(2 (3)), 136–153.Google Scholar
Kac, M. 1966. Can one hear the shape of a drum? Amer. Math. Monthly, 73(4, part II), 1–23.Google Scholar
Kalf, H., Schmincke, U.-W., Walter, J., and Wüst, R. 1975. On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials. Pages 182–226 of: Spectral Theory and Differential Equations, (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens). Everitt, W. N. (editor). Lecture Notes in Math., Vol. 448. Springer-Verlag, Berlin.Google Scholar
Karpukhin, M., Nadirashvili, N., Penskoi, A. V., and Polterovich, I. 2021. An isoperimetric inequality for Laplace eigenvalues on the sphere. J. Differential Geom., 118(2), 313–333.Google Scholar
Kashin, B. S. 2006. On a class of inequalities for orthonormal systems. Mat. Zametki, 80(2), 204–208. English translation in Math. Notes, 80(1–2), 199–203, (2006).Google Scholar
Kato, T. 1951. Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Amer. Math. Soc., 70, 195–211.Google Scholar
Kato, T. 1972. Schrödinger operators with singular potentials. Israel J. Math., 13, 135–148 (1973).CrossRefGoogle Scholar
Kato, T. 1978. Remarks on Schrödinger operators with vector potentials. Integral Equations Operator Theory, 1(1), 103–113.Google Scholar
Kato, T. 1980. Perturbation Theory for Linear Operators. Reprinted 1995 in Classics in Mathematics. Springer-Verlag, Berlin.Google Scholar
Kay, I., and Moses, H. E. 1956. Reflectionless transmission through dielectrics and scattering potentials. J. Appl. Phys., 27(12), 1503–1508.Google Scholar
Keller, J. B. 1961. Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation. J. Math. Phys., 2, 262–266.Google Scholar
Kellner, R. 1966. On a theorem of Pólya. Amer. Math. Monthly, 73, 856–858.Google Scholar
Kenig, C. E., Ruiz, A., and Sogge, C. D. 1987. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J., 55(2), 329–347.Google Scholar
Khuri, N. N., Martin, A., and Wu, T. T. 2002. Bound states in n dimensions (especially n = 1 and n = 2). Few-Body Systems , 31, 83–89.CrossRefGoogle Scholar
Killip, R. 2007. Spectral theory via sum rules. Pages 907–930 of: Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday. Gesztesy, F., Deift, P., Galvez, C., Perry, P., and Schlag, W. (editors). Proc. Sympos. Pure Math., vol. 76. Amer. Math. Soc., Providence, RI.CrossRefGoogle Scholar
Killip, R., and Simon, B. 2003. Sum rules for Jacobi matrices and their applications to spectral theory. Ann. of Math. (2), 158(1), 253–321.Google Scholar
Killip, R., and Simon, B. 2009. Sum rules and spectral measures of Schrödinger operators with L2 potentials. Ann. of Math. (2), 170(2), 739–782.Google Scholar
Killip, R., Vişan, M., and Zhang, X. 2018. Low regularity conservation laws for integrable PDE. Geom. Funct. Anal., 28(4), 1062–1090.Google Scholar
Klaus, M. 1977. On the bound state of Schrödinger operators in one dimension. Ann. Phys., 108(2), 288–300.CrossRefGoogle Scholar
Klaus, M., and Simon, B. 1980. Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case. Ann. Physics, 130(2), 251–281.Google Scholar
Klaus, M. 1982/83. Some applications of the Birman–Schwinger principle. Helv. Phys. Acta, 55(1), 49–68.Google Scholar
Kondrachov, W. 1945. Sur certaines propriétés des fonctions dans l’espace. C. R. (Dokl.) Acad. Sci. URSS, n. Ser., 48, 535–538.Google Scholar
Korevaar, N. 1993. Upper bounds for eigenvalues of conformal metrics. J. Diff. Geom., 37(1), 73–93.Google Scholar
Kovařík, H., and Weidl, T. 2015. Improved Berezin–Li–Yau inequalities with magnetic field. Proc. Roy. Soc. Edinburgh Sect. A, 145(1), 145–160.Google Scholar
Kovařík, H., Vugalter, S., and Weidl, T. 2007. Spectral estimates for two-dimensional Schrödinger operators with application to quantum layers. Comm. Math. Phys., 275(3), 827–838.Google Scholar
Kovařík, H., Vugalter, S., and Weidl, T. 2009. Two-dimensional Berezin–Li–Yau inequalities with a correction term. Comm. Math. Phys., 287(3), 959–981.Google Scholar
Krahn, E. 1925. Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann., 94, 97–100.CrossRefGoogle Scholar
Krahn, E. 1926. Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen. Acta Univ. Dorpat A, 9, 1–44. English translation in Edgar Krahn 1894–1961. A Centenary Volume. Ü. Lumiste and J. Peetre (editors). Amsterdam: IOS Press; Tartu: The Estonian Mathematical Society, 139–175, (1994).Google Scholar
Kreĭn, M. G. 1951. Determination of the density of a nonhomogeneous symmetric cord by its frequency spectrum. Doklady Akad. Nauk SSSR (N.S.), 76, 345–348.Google Scholar
Kröger, P. 1992. Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space. J. Funct. Anal., 106(2), 353–357.Google Scholar
Kufner, A., Maligranda, L., and Persson, L.-E. 2006. The prehistory of the Hardy inequality. Amer. Math. Monthly, 113(8), 715–732.CrossRefGoogle Scholar
Kufner, A., Maligranda, L., and Persson, L.-E. 2007. The Hardy Inequality. About its History and Some Related Results. Vydavatelský Servis, Plzeň.Google Scholar
Kuznetsov, N. V., and Fedosov, B. V. 1967. Asymptotische Formel für die Eigenwerte einer Kreismembran. Differ. Equation, 1, 1326–1329.Google Scholar
Kwaśnicki, M., Laugesen, R. S., and Siudeja, B. A. 2019. Pólya’s conjecture fails for the fractional Laplacian. J. Spectr. Theory, 9(1), 127–135.Google Scholar
Kwong, M. K. 1989. Uniqueness of positive solutions of Δu - u + up = 0 in Rn. Arch. Rational Mech. Anal., 105(3), 243–266.Google Scholar
Lamé, G. 1833. Mémoire sur la propagation de la chaleur dans les polyèdres. J. Éc. Polytec. Math., 22, 194–251.Google Scholar
Landau, L. D., and Lifshitz, E. M. 1958. Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Vol. 3. Translated from the Russian by Sykes, J. B. and Bell, J. S.. Pergamon Press Ltd., London and Paris; and Addison-Wesley Publishing Co., Inc., Reading, MA.Google Scholar
Langmann, E., Laptev, A., and Paufler, C. 2006. Singular factorizations, self-adjoint extensions and applications to quantum many-body physics. J. Phys. A, 39(5), 1057–1071.Google Scholar
Lapidus, M. L., and Pomerance, C. 1993. The Riemann zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums. Proc. Lond. Math. Soc. (3), 66(1), 41–69.Google Scholar
Lapidus, M. L., and Pomerance, C. 1996. Counterexamples to the modified Weyl–Berry conjecture on fractal drums. Math. Proc. Camb. Philos. Soc., 119(1), 167–178.Google Scholar
Laptev, A. A. 1981. Spectral asymptotics of a class of Fourier integral operators. Trudy Moskov. Mat. Obshch., 43, 92–115. English translation in Trans. Mosc. Math. Soc, 1, 101–127 (1983).Google Scholar
Laptev, A. 1993. Asymptotics of the negative discrete spectrum of a class of Schrödinger operators with large coupling constant. Proc. Amer. Math. Soc., 119(2), 481–488.Google Scholar
Laptev, A. 1995. On inequalities for the bound states of Schrödinger operators. Pages 221–225 of: Partial Differential Operators and Mathematical Physics (Holzhau, 1994). Demuth, M. and Schulze, B.-W. (editors). Oper. Theory Adv. Appl., vol. 78. Birkhäuser, Basel.Google Scholar
Laptev, A. 1997. Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J. Funct. Anal., 151(2), 531–545.Google Scholar
Laptev, A. 1999. On the Lieb–Thirring conjecture for a class of potentials. Pages 227–234 of: The Maz’ya Anniversary Collection, Vol. 2 (Rostock, 1998). Rossmann, J., Takáč, P., and Wildenhain, G. (editors). Oper. Theory Adv. Appl., vol.110. Birkhäuser, Basel.Google Scholar
Laptev, A. 2021. On factorisation of a class of Schrödinger operators. Complex Var. Elliptic Equ., 66(6–7), 1100–1107.Google Scholar
Laptev, A., and Safarov, Yu. 1996. A generalization of the Berezin–Lieb inequality. Pages 69–79 of: Contemporary Mathematical Physics. Minlos, R. A., Shubin, M. A., and Vershik, A. M. (editors). Amer. Math. Soc. Transl. Ser. 2, vol. 175. Amer. Math. Soc., Providence, RI.Google Scholar
Laptev, A., and Safronov, O. 2009. Eigenvalue estimates for Schrödinger operators with complex potentials. Comm. Math. Phys., 292(1), 29–54.CrossRefGoogle Scholar
Laptev, A., and Solomyak, M. 2012. On the negative spectrum of the two-dimensional Schrödinger operator with radial potential. Comm. Math. Phys., 314(1), 229–241.Google Scholar
Laptev, A., and Solomyak, M. 2013. On spectral estimates for two-dimensional Schrödinger operators. J. Spectr. Theory, 3(4), 505–515.Google Scholar
Laptev, A., and Weidl, T. 2000a. Recent results on Lieb–Thirring inequalities. Exp. No. XX of: Journ. Équ. Dériv. Partielles (La Chapelle sur Erdre, 2000). Univ. Nantes, Nantes.Google Scholar
Laptev, A., and Weidl, T. 2000b. Sharp Lieb–Thirring inequalities in high dimensions. Acta Math., 184(1), 87–111.Google Scholar
Laptev, A., Naboko, S., and Safronov, O. 2003. On new relations between spectral properties of Jacobi matrices and their coefficients. Comm. Math. Phys., 241(1), 91–110.Google Scholar
Laptev, A., Naboko, S., and Safronov, O. 2005. Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials. Comm. Math. Phys., 253(3), 611–631.Google Scholar
Laptev, A., Loss, M., and Schimmer, L. 2021. A remark on a paper by Hundertmark and Simon. Preprint, available at ArXiv:2012.13793.Google Scholar
Larson, S. 2017. On the remainder term of the Berezin inequality on a convex domain. Proc. Amer. Math. Soc., 145(5), 2167–2181.Google Scholar
Larson, S., Lundholm, D., and Nam, P. T. 2021. Lieb–Thirring inequalities for wave functions vanishing on the diagonal set. Ann. H. Lebesgue, 4, 251–282.Google Scholar
Latushkin, Y., and Sukhtayev, A. 2010. The algebraic multiplicity of eigenvalues and the Evans function revisited. Math. Model. Nat. Phenom., 5(4), 269–292.Google Scholar
Lax, P. D. 1968. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 21, 467–490.Google Scholar
Lax, P. D. 2002. Functional Analysis. John Wiley & Sons, New York.Google Scholar
Leinfelder, H., and Simader, C. G. 1981. Schrödinger operators with singular magnetic vector potentials. Math. Z., 176(1), 1–19.Google Scholar
Lenard, A., and Dyson, F. J. 1968. Stability of matter. II. J. Math. Phys., 9(5), 698–711.Google Scholar
Leoni, G. 2017. A First Course in Sobolev Spaces. Second edn. Graduate Studies in Mathematics, vol. 181. Amer. Math. Soc., Providence, RI.Google Scholar
Leoni, G., and Morini, M. 2007. Necessary and sufficient conditions for the chain rule in (ℝN;ℝd) and BVloc(ℝN;ℝd). J. Eur. Math. Soc. (JEMS), 9(2), 219–252.Google Scholar
Levin, D., and Solomyak, M. 1997. The Rozenblum–Lieb–Cwikel inequality for Markov generators. J. Anal. Math., 71, 173–193.Google Scholar
Levine, H. A., and Weinberger, H. F. 1986. Inequalities between Dirichlet and Neumann eigenvalues. Arch. Rational Mech. Anal., 94(3), 193–208.Google Scholar
Levinson, N. 1949. On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase. Danske Vid. Selsk. Mat.-Fys. Medd., 25(9), 29.Google Scholar
Levitan, B. M. 1952. On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order. Izvestiya Akad. Nauk SSSR. Ser. Mat., 16, 325–352.Google Scholar
Levitin, M., and Vassiliev, D. 1996. Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals. Proc. Lond. Math. Soc. (3), 72(1), 188–214.Google Scholar
Levitin, M., Polterovich, I., and Sher, D. A. 2022. Pólya’s conjecture for the disk: a computer-assisted proof. Preprint, available at ArXiv.2203.07696.Google Scholar
Lewin, M. 2022. Théorie Spectrale et Mécanique Quantique. Mathématiques et Applications. Springer.Google Scholar
Lewin, M., and Sabin, J. 2014. The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D. Anal. PDE, 7(6), 1339–1363.Google Scholar
Lewin, M., and Sabin, J. 2015. The Hartree equation for infinitely many particles I. Well-posedness theory. Comm. Math. Phys., 334(1), 117–170.CrossRefGoogle Scholar
Lewin, M., Lieb, E. H., and Seiringer, R. 2018. Statistical mechanics of the uniform electron gas. J. Éc. Polytech. Math., 5, 79–116.Google Scholar
Lewin, M., Lieb, E. H., and Seiringer, R. 2020a. The local density approximation in density functional theory. Pure Appl. Anal., 2(1), 35–73.Google Scholar
Lewin, M., Lieb, E. H., and Seiringer, R. 2020b. Universal functionals in density functional theory. Preprint, available at ArXiv:1912.10424.Google Scholar
Li, P., and Yau, S. T. 1980. Estimates of eigenvalues of a compact Riemannian manifold. Pages 205–239 of: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979). Osserman, R. and Weinstein, A. (editors). Proc. Sympos. Pure Math., XXXVI. Amer. Math. Soc., Providence, RI.Google Scholar
Li, P., and Yau, S. T. 1983. On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys., 88(3), 309–318.Google Scholar
Lieb, E. H. 1976. Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc., 82(5), 751–753.Google Scholar
Lieb, E. H. 1973. The classical limit of quantum spin systems. Comm. Math. Phys., 31, 327–340.Google Scholar
Lieb, E. H. 1976/77. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies in Appl. Math., 57(2), 93–105.Google Scholar
Lieb, E. H. 1980. The number of bound states of one-body Schrödinger operators and the Weyl problem. Pages 241–252 of: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979). Osserman, R. and Weinstein, A. (editors). Proc. Sympos. Pure Math., XXXVI. Amer. Math. Soc., Providence, RI.Google Scholar
Lieb, E. H. 1981. Thomas–Fermi and related theories of atoms and molecules. Rev. Modern Phys., 53(4), 603–641. Erratum ibid. 54(1), p. 311 (1982).Google Scholar
Lieb, E. H. 1983a. Density functionals for Coulomb systems. Int. J. Quantum Chem., 24(3), 243–277.Google Scholar
Lieb, E. H. 1983b. An Lp bound for the Riesz and Bessel potentials of orthonormal functions. J. Funct. Anal., 51(2), 159–165.CrossRefGoogle Scholar
Lieb, E. H. 1983c. On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math., 74(3), 441–448.Google Scholar
Lieb, E. H. 1983d. Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. of Math. (2), 118(2), 349–374.Google Scholar
Lieb, E. H. 1984. On characteristic exponents in turbulence. Comm. Math. Phys., 92(4), 473–480.Google Scholar
Lieb, E. H., and Loss, M. 2001. Analysis. Second edn. Graduate Studies in Mathematics, vol. 14. Amer. Math. Soc., Providence, RI.Google Scholar
Lieb, E. H., and Seiringer, R. 2010. The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge.Google Scholar
Lieb, E. H., and Simon, B. 1977. The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math., 23(1), 22–116.Google Scholar
Lieb, E. H., and Thirring, W. E. 1975. Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett., 35(Sep), 687–689.Google Scholar
Lieb, E. H., and Thirring, W. E. 1976. Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Pages 269–303 of: Studies in Mathematical Physics (Essays in Honor of Valentine Bargmann). Lieb, E. H., Simon, B., and Wightman, A. S. (editors). Princeton University Press, Princeton, NJ.Google Scholar
Lieb, E. H., Siedentop, H., and Solovej, J. P. 1997. Stability and instability of relativistic electrons in classical electromagnetic fields. J. Stat. Phys., 89, 37–59.Google Scholar
Linde, H. 2006. A lower bound for the ground state energy of a Schrödinger operator on a loop. Proc. Amer. Math. Soc., 134(12), 3629–3635.Google Scholar
Lions, P.-L. 1984a. The concentration–compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1(2), 109–145.Google Scholar
Lions, P.-L. 1984b. The concentration–compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1(4), 223–283.Google Scholar
Lions, P.-L., and Paul, T. 1993. Sur les mesures de Wigner. Rev. Mat. Iberoamericana, 9(3), 553–618.Google Scholar
Lorentz, H. A. 1910. Alte und neue Fragen der Physik. Phys. Z., 11, 1234–1257.Google Scholar
Lundholm, D., and Seiringer, R. 2018. Fermionic behavior of ideal anyons. Lett. Math. Phys., 108(11), 2523–2541.Google Scholar
Lundholm, D., and Solovej, J. P. 2013. Hardy and Lieb–Thirring inequalities for anyons. Comm. Math. Phys., 322(3), 883–908.Google Scholar
Lundholm, D., and Solovej, J. P. 2014. Local exclusion and Lieb–Thirring inequalities for intermediate and fractional statistics. Ann. Henri Poincaré, 15(6), 1061–1107.Google Scholar
Lundholm, D., Portmann, F., and Solovej, J. P. 2015. Lieb–Thirring bounds for interacting Bose gases. Comm. Math. Phys., 335(2), 1019–1056.Google Scholar
Lundholm, D., Nam, P. T., and Portmann, F. 2016. Fractional Hardy–Lieb–Thirring and related inequalities for interacting systems. Arch. Rational Mech. Anal., 219(3), 1343–1382.Google Scholar
Makai, E. 1965. A lower estimation of the principal frequencies of simply connected membranes. Acta Math. Acad. Sci. Hungar., 16, 319–323.Google Scholar
Marchenko, V. A. 2011. Sturm–Liouville Operators and Applications. Revised edn. AMS Chelsea Publishing, Providence, RI.Google Scholar
Martin, A. 1972. Bound states in the strong coupling limit. Helv. Phys. Acta, 45, 140–148.Google Scholar
Martin, A. 1990. New results on the moments of the eigenvalues of the Schrödinger Hamiltonian and applications. Comm. Math. Phys., 129(1), 161–168.Google Scholar
Matveev, V. B., and Salle, M. A. 1991. Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer-Verlag, Berlin.Google Scholar
Maz’ya, V. G. 1960. Classes of domains and imbedding theorems for function spaces. Sov. Math., Dokl., 1, 882–885.Google Scholar
Maz’ya, V. G. 1962. The negative spectrum of the higher-dimensional Schrödinger operator. Dokl. Akad. Nauk SSSR, 144, 721–722.Google Scholar
Maz’ya, V. G. 1964. On the theory of the higher-dimensional Schrödinger operator. Izv. Akad. Nauk SSSR Ser. Mat., 28, 1145–1172.Google Scholar
Maz’ya, V. G. 1969. On Neumann’s problem in domains with nonregular boundaries. Sib. Math. J., 9, 990–1012.Google Scholar
Maz’ya, V. G. 1974. On connection of two kinds of capacity. Vestn. Leningr. Univ., Mat. Mekh. Astron., 1974(2), 33–40.Google Scholar
Maz’ya, V. 2003. Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Pages 307–340 of: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002). Auscher, P., Coulhon, T., and Grigoryan, A. (editors). Contemp. Math., vol. 338. Amer. Math. Soc., Providence, RI.Google Scholar
Maz’ya, V. 2007. Analytic criteria in the qualitative spectral analysis of the Schrödinger operator. Pages 257–288 of: Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday. Gesztesy, F., Deift, P., Galvez, C., Perry, P., and Schlag, W. (editors). Proc. Sympos. Pure Math., vol. 76. Amer. Math. Soc., Providence, RI.Google Scholar
Maz’ya, V. 2011. Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Augmented edn. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer-Verlag, Heidelberg.Google Scholar
Maz’ya, V. G., and Otelbaev, M. 1977. Imbedding theorems and the spectrum of a certain pseudodifferential operator. Sibirsk. Mat. Ž., 18(5), 1073–1087, 1206. English translation in Siberian Math. J., 18(5), 758–770, (1977).Google Scholar
Maz’ya, V., and Shubin, M. 2005a. Can one see the fundamental frequency of a drum? Lett. Math. Phys., 74(2), 135–151.Google Scholar
Maz’ya, V., and Shubin, M. 2005b. Discreteness of spectrum and positivity criteria for Schrödinger operators. Ann. of Math. (2), 162(2), 919–942.Google Scholar
Maz’ya, V. G., and Verbitsky, I. E. 2002. The Schrödinger operator on the energy space: boundedness and compactness criteria. Acta Math., 188(2), 263–302.Google Scholar
Maz’ya, V. G., and Verbitsky, I. E. 2005. Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator. Invent. Math., 162(1), 81–136.Google Scholar
McLeod, J. B. 1961. The distribution of the eigenvalues for the hydrogen atom and similar cases. Proc. Lond. Math. Soc. (3), 11, 139–158.Google Scholar
McLeod, K. 1993. Uniqueness of positive radial solutions of Δu + f (u) = 0inRn . II. Trans. Amer. Math. Soc., 339(2), 495–505.Google Scholar
McLeod, K., and Serrin, J. 1987. Uniqueness of positive radial solutions of Δu + f (u) = 0 in Rn. Arch. Rational Mech. Anal., 99(2), 115–145.Google Scholar
Melas, A. D. 2003. A lower bound for sums of eigenvalues of the Laplacian. Proc. Amer. Math. Soc., 131(2), 631–636.Google Scholar
Melrose, R. B. 1980. Weyl’s conjecture for manifolds with concave boundary. Pages 257–274 of: Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979). Osserman, R. and Weinstein, A. (editors). Proc. Sympos. Pure Math., XXXVI. Amer. Math. Soc., Providence, RI.Google Scholar
Metivier, G. 1982. Estimation du reste en théorie spectrale. Journ. Équ. Dériv. Partielles, Saint-Jean-De-Monts 1982, Exp. No. 1, 5 p. http://eudml.org/doc/93075.Google Scholar
Meyers, N.G., and Serrin, J. 1964. H = W. Proc. Nat. Acad. Sci. USA., 51, 1055–1056.Google Scholar
Mikhaĭlov, V. P. 1978. Partial Differential Equations. Translated from the Russian by Sinha, P. C.. “MIR”, Moscow; distributed by Imported Publications, Inc., Chicago, Ill.Google Scholar
Milnor, J. 1964. Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Acad. Sci. U.S.A., 51, 542.Google Scholar
Minakshisundaram, S., and Pleijel, Å. 1949. Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canad. J. Math., 1, 242–256.Google Scholar
Mitrinović, D. S. 1970. Analytic Inequalities. In cooperation with P. M. Vasić. Grundlehren der Mathematischen Wissenschaften, Band 165. Springer-Verlag, New York and Berlin.Google Scholar
Miura, R. M., Gardner, C. S., and Kruskal, M. D. 1968. Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys., 9, 1204–1209.Google Scholar
Molčanov, A. M. 1953. On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order. Trudy Moskov. Mat. Obšč., 2, 169–199.Google Scholar
Molchanov, S., and Vainberg, B. 1997. On spectral asymptotics for domains with fractal boundaries. Comm. Math. Phys., 183(1), 85–117.Google Scholar
Molchanov, S., and Vainberg, B. 1998. On spectral asymptotics for domains with fractal boundaries of cabbage type. Math. Phys. Anal. Geom., 1(2), 145–170.Google Scholar
Molchanov, S., and Vainberg, B. 2010. On general Cwikel–Lieb–Rozenblum and Lieb–Thirring inequalities. Pages 201–246 of: Around the Research of Vladimir Maz’ya. III. Analysis and Applications. Laptev, A. (editor). Springer, Dordrecht; Tamara Rozhkovskaya Publisher, Novosibirsk.Google Scholar
Molchanov, S., and Vainberg, B. 2012. Bargmann type estimates of the counting function for general Schrödinger operators. J. Math. Sciences, 184(4), 457–508.Google Scholar
Molchanov, S., Novitskii, M., and Vainberg, B. 2001. First KdV integrals and absolutely continuous spectrum for 1-D Schrödinger operator. Comm. Math. Phys., 216(1), 195–213.Google Scholar
Montiel, S„ and Ros, A. 1986. Minimalimmersionsofsurfacesbythefirsteigenfunctions and conformal area. Invent. Math., 83(1), 153–166.Google Scholar
Morpurgo, C. 1999. Sharp trace inequalities for intertwining operators on Sn and Rn. Internat. Math. Res. Notices, 1999(20), 1101–1117.Google Scholar
Morpurgo, C. 2002. Sharp inequalities for functional integrals and traces of conformally invariant operators. Duke Math. J., 114(3), 477–553.Google Scholar
Moser, T., and Seiringer, R. 2017. Stability of a fermionic N + 1 particle system with point interactions. Comm. Math. Phys., 356(1), 329–355.Google Scholar
Muckenhoupt, B. 1972. Hardy’s inequality with weights. Studia Math., 44, 31–38.Google Scholar
Naimark, K., and Solomyak, M. 1997. Regular and pathological eigenvalue behavior for the equation -λu″ = Vu on the semiaxis. J. Funct. Anal., 151(2), 504–530.CrossRefGoogle Scholar
Nakamura, S. 2020. The orthonormal Strichartz inequality on torus. Trans. Amer. Math. Soc., 373(2), 1455–1476.Google Scholar
Nam, P. T. 2018. Lieb–Thirring inequality with semiclassical constant and gradient error term. J. Funct. Anal., 274(6), 1739–1746.Google Scholar
Nam, P. T. 2022. A proof of the Lieb–Thirring inequality via the Besicovitch covering lemma. Preprint, available at ArXiv:2206.15368.Google Scholar
Naumann, J. 2002. Remarks on the Prehistory of Sobolev Spaces. Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik.Google Scholar
Netrusov, Y., and Weidl, T. 1996. On Lieb–Thirring inequalities for higher order operators with critical and subcritical powers. Comm. Math. Phys., 182(2), 355–370.Google Scholar
Netrusov, Yu., and Safarov, Yu. 2005. Weyl asymptotic formula for the Laplacian on domains with rough boundaries. Comm. Math. Phys., 253(2), 481–509.Google Scholar
Nirenberg, L. 1959. On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser., 13, 115–162.Google Scholar
Opic, B., and Kufner, A. 1990. Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series, vol. 219. Longman Scientific & Technical, Harlow.Google Scholar
Pauli, W. 1926. Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik. Z. Phys., 36(5), 336–363.Google Scholar
Pavlov, B. S. 1966. On a non-selfadjoint Schrödinger operator. Pages 102–132 of: Spectral Theory and Wave Processes. Problems of Mathematical Physics, No. 1. Izdat. Leningrad. Univ., Leningrad.Google Scholar
Pavlov, B. S. 1967. On a non-selfadjoint Schrödinger operator. II. Pages 133–157 of: Spectral Theory, Diffraction Problems. Problems of Mathematical Physics, No. 2. Izdat. Leningrad. Univ., Leningrad.Google Scholar
Pavlović, N. 2003. Bounds for sums of powers of eigenvalues of Schrödinger operators via the commutation method. Pages 271–281 of: Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002). Karpeshina, Y., Stolz, G., Weikard, R., and Zeng, Y. (editors). Contemp. Math., vol. 327. Amer. Math. Soc., Providence, RI.Google Scholar
Payne, L. E. 1955. Inequalities for eigenvalues of membranes and plates. J. Rational Mech. Anal., 4, 517–529.Google Scholar
Payne, L. E., and Stakgold, I. 1973. On the mean value of the fundamental mode in the fixed membrane problem. Applicable Anal., 3, 295–306.Google Scholar
Payne, L. E., and Weinberger, H. F. 1960. An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal., 5, 286–292 (1960).Google Scholar
Petkov, V. M., and Stojanov, L. N. 1988. On the number of periodic reflecting rays in generic domains. Ergodic Theory Dynam. Systems, 8(1), 81–91.Google Scholar
Pinsky, M. A. 1980. The eigenvalues of an equilateral triangle. SIAM J. Math. Anal., 11(5), 819–827.Google Scholar
Pinsky, M. A. 1985. Completeness of the eigenfunctions of the equilateral triangle. SIAM J. Math. Anal., 16(4), 848–851.Google Scholar
Pockels, F. C. A. 1891. Über die Partielle Differentialgleichung Δu+k2u = 0 und deren Auftreten in der Mathematischen Physik. B. G. Teubner, Leipzig.Google Scholar
Poincaré, H. 1890. Sur les équations aux dérivées partielles de la physique mathématique. Amer. J. Math., 12(3), 211–294.Google Scholar
Poincaré, H. 1894. Sur les équations de la physique mathématique. Rend. Circ. Mat. Palermo, 8, 57–156.CrossRefGoogle Scholar
Poincaré, H. 1897. La méthode de Neumann et le problème de Dirichlet. Acta Math., 20, 59–142.Google Scholar
Pólya, G. 1952. Remarks on the foregoing paper. J. Math. Phys., 31, 55–57.Google Scholar
Pólya, G. 1954. Patterns of Plausible Inference. Mathematics and Plausible Reasoning, vol. II. Princeton University Press, Princeton, NJ.Google Scholar
Pólya, G. 1955. On the characteristic frequencies of a symmetric membrane. Math. Z., 63, 331–337.Google Scholar
Pólya, G. 1961. On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. (3), 11, 419–433.Google Scholar
Pöschl, G., and Teller, E. 1933. Bemerkungen zur Quantenmechanik des anharmonischen Oszillators. Z. Phys., 83, 143–151.Google Scholar
Protter, M. H. 1981. A lower bound for the fundamental frequency of a convex region. Proc. Amer. Math. Soc., 81(1), 65–70.Google Scholar
Pushnitski, A. 2009. Operator theoretic methods for the eigenvalue counting function in spectral gaps. Ann. Henri Poincaré, 10(4), 793–822.Google Scholar
Pushnitski, A. 2011. The Birman–Schwinger principle on the essential spectrum. J. Funct. Anal., 261(7), 2053–2081.Google Scholar
Radin, C. 1994. The pinwheel tilings of the plane. Ann. of Math. (2), 139(3), 661–702.Google Scholar
Rayleigh, John William Strutt, Lord. 1877. The Theory of Sound. MacMillan and Co. London.Google Scholar
Reed, M., and Simon, B. 1972. Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York and London.Google Scholar
Reed, M., and Simon, B. 1975. Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York and London.Google Scholar
Reed, M., and Simon, B. 1978. Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York and London.Google Scholar
Reed, M., and Simon, B. 1979. Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press, New York and London.Google Scholar
Rellich, F. 1930. Ein Satz über mittlere Konvergenz. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 1930, 30–35.Google Scholar
Rellich, F. 1942. Störungstheorie der Spektralzerlegung. V. Math. Ann., 118, 462–484.Google Scholar
Ritz, W. 1908. Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. Reine Angew. Math., 135, 1–61.Google Scholar
Robinson, D. W. 1971. The Thermodynamic Pressure in Quantum Statistical Mechanics. Lecture Notes in Physics, Vol. 9. Springer-Verlag, Berlin and New York.Google Scholar
Rodemich, E. 1966. The Sobolev inequalities with best possible constants. Analysis Seminar at California Institute of Technology, 1–25.Google Scholar
Rogers, L. G. 2006. Degree-independent Sobolev extension on locally uniform domains. J. Funct. Anal., 235(2), 619–665.Google Scholar
Rosen, G. 1971. Minimum value for c in the Sobolev inequality ||ϕ3 || ≤ c||∇ϕ||3. SIAM J. Appl. Math., 21, 30–32.Google Scholar
Rotfel’d, S. Ju. 1967. Remarks on the singular values of a sum of completely continuous operators. Funkcional. Anal. i Priložen, 1(3), 95–96. English translation in Functional Analysis and its Applications, 1(3), 252–253 (1967).Google Scholar
Rotfel’d, S. Ju. 1968. The singular values of the sum of completely continuous operators. Pages 81–87 of: Spectral Theory. Birman, M. Sh. (editor). Problems of Mathematical Physics, No. 3 Izdat. Leningrad. Univ., Leningrad. English translation in Spectral Theory. Topics in Mathematical Physics 3, 73–78 (1969). Consultants Bureau, New York and London.Google Scholar
Royden, H. L. 1963. Real Analysis. The Macmillan Co., New York; Collier–Macmillan Ltd., London.Google Scholar
Rozenbljum, G. V. 1971. The distribution of the eigenvalues of the first boundary value problem in unbounded domains. Dokl. Akad. Nauk SSSR, 200, 1034–1036. English translation in Soviet Math. Dokl., 12, 1539–1542, (1971).Google Scholar
Rozenbljum, G. V. 1972a. Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR, 202, 1012–1015. English translation in Soviet Math. Dokl., 13, 245–249, (1972).Google Scholar
Rozenbljum, G. V. 1972b. The eigenvalues of the first boundary value problem in unbounded domains. Mat. Sb. (N.S.), 89 (131), 234–247, 350. English translation in Math USSR Sb., 18(2), 235–248, (1972).Google Scholar
Rozenbljum, G. V. 1973. The calculation of the spectral asymptotics for the Laplace operator in domains of infinite measure. Pages 95–106, 144 of: Problems of Mathematical Analysis, No. 4: Integral and Differential Operators. Differential Equations, English translation in J. Soviet Math., 6, 64–71, (1976).Google Scholar
Rozenbljum, G. V. 1976. Distribution of the discrete spectrum of singular differential operators. Izv. Vysš. Učebn. Zaved. Matematika, 1, 75–86. English translation in Soviet Math. (Iz. VUZ), 20(1), 63–71, (1976).Google Scholar
Rozenblum, G. 2021. Lieb–Thirring estimates for singular measures. Preprint, available at ArXiv:2108.11429.Google Scholar
Rozenblum, G., and Shargorodsky, E. 2021. Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case. Preprint, available at ArXiv:2011.14877.Google Scholar
Rozenblyum, G., and Solomyak, M. 1997. The Cwikel–Lieb–Rozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups. Algebra i Analiz, 9(6), 214–236. English translation in St. Petersburg Math. J., 9(6), 1195–1211 (1998).Google Scholar
Rozenblum, G., and Tashchiyan, G. 2021. Eigenvalues of the Birman–Schwinger operator for singular measures: the noncritical case. Preprint, available at ArXiv:2107.04682.Google Scholar
Rozenblum, G. V., Shubin, M. A., and Solomyak, M. Z. 1989. Partial Differential Equations VII: Spectral Theory of Differential Operators. Encyclopedia of Mathematical Sciences. vol. 64. Translated from the Russian by T. Zastawniak. Springer-Verlag, Berlin.Google Scholar
Rudin, W. 1987. Real and Complex Analysis. Third edn. McGraw-Hill Book Co., New York.Google Scholar
Rudin, W. 1991. Functional Analysis. Second edn. McGraw-Hill, Inc., New York.Google Scholar
Rumin, M. 2011. Balanced distribution-energy inequalities and related entropy bounds. Duke Math. J., 160(3), 567–597.Google Scholar
Sabin, J. 2016. Littlewood–Paley decomposition of operator densities and application to a new proof of the Lieb–Thirring inequality. Math. Phys. Anal. Geom., 19(2), Art. 11, 11.Google Scholar
Safarov, Yu., and Vassiliev, D. 1997. The Asymptotic Distribution of Eigenvalues of Partial Differential Operators. Translations of Mathematical Monographs, vol. 155. Amer. Math. Soc., Providence, RI.Google Scholar
Safronov, O. L. 1996. The discrete spectrum in gaps of the continuous spectrum for indefinite-sign perturbations with a large coupling constant. Algebra i Analiz, 8(2), 162–194. English translation in St. Petersburg Math. J., 8(2), 307–331, (1997).Google Scholar
Safronov, O. 2005. On the absolutely continuous spectrum of multi-dimensional Schrödinger operators with slowly decaying potentials. Comm. Math. Phys., 254(2), 361–366.Google Scholar
Schimmer, L. 2015. Spectral inequalities for Jacobi operators and related sharp Lieb–Thirring inequalities on the continuum. Comm. Math. Phys., 334(1), 473–505.Google Scholar
Schmetzer, T. 2022. New interpolation results for Lieb–Thirring inequalities. M.Sc thesis, University of Stuttgart. Available at https://codeberg.org/attachments/a1a0d333-e7fb-400f-a7d3-a0345161a8f8.Google Scholar
Schmincke, U.-W. 1978. On Schrödinger’s factorization method for Sturm–Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A, 80(1–2), 67–84.Google Scholar
Schmincke, U.-W. 2003. On a paper by F. Gesztesy, B. Simon, and G. Teschl concerning isospectral deformations of ordinary Schrödinger operators: “Spectral deformations of one-dimensional Schrödinger operators” [J. Anal. Math. 70 (1996), 267–324; MR1444263 (98m:34171)]. J. Math. Anal. Appl., 277(1), 51–78.Google Scholar
Schrödinger, E. 1926a. Quantisierung als Eigenwertproblem. Ann. der Phys., 384(4), 361–376.Google Scholar
Schrödinger, E. 1926b. Quantisierung als Eigenwertproblem. II. Ann. der Phys., 79(4), 489–527.Google Scholar
Schwartz, L. 1950. Théorie des Distributions. Tome I. Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1091. Hermann & Cie., Paris.Google Scholar
Schwartz, L. 1951. Théorie des Distributions. Tome II. Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1122. Hermann & Cie., Paris.Google Scholar
Schwinger, J. 1961. On the bound states of a given potential. Proc. Nat. Acad. Sci. USA., 47, 122–129.Google Scholar
Seeley, R. 1978. A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of R3. Adv. Math., 29, 244–269.Google Scholar
Seeley, R. 1980. An estimate near the boundary for the spectral function of the Laplace operator. Amer. J. Math., 102(5), 869–902.Google Scholar
Seiler, E., and Simon, B. 1975. Bounds in the Yukawa2 quantum field theory: upper bound on the pressure, Hamiltonian bound and linear lower bound. Comm. Math. Phys., 45(2), 99–114.Google Scholar
Serrin, J., and Tang, M. 2000. Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J., 49(3), 897–923.Google Scholar
Shargorodsky, E. 2013. An estimate for the Morse index of a Stokes wave. Arch. Rational Mech. Anal., 209(1), 41–59.CrossRefGoogle Scholar
Shargorodsky, E. 2014. On negative eigenvalues of two-dimensional Schrödinger operators. Proc. Lond. Math. Soc. (3), 108(2), 441–483.Google Scholar
Shubin, M. A. 2001. Pseudodifferential Operators and Spectral Theory. Second edn. Translated from the 1978 Russian original by Andersson, S. I.. Springer-Verlag, Berlin.Google Scholar
Sigal, I. M. 1983. Geometric parametrices and the many-body Birman–Schwinger principle. Duke Math. J., 50(2), 517–537.Google Scholar
Simon, B. 1976a. Analysis with weak trace ideals and the number of bound states of Schrödinger operators. Trans. Amer. Math. Soc., 224(2), 367–380.Google Scholar
Simon, B. 1976b. The bound state of weakly coupled Schrödinger operators in one and two dimensions. Ann. Physics, 97(2), 279–288.Google Scholar
Simon, B. 1979a. Functional Integration and Quantum Physics. Pure and Applied Mathematics, vol. 86. Academic Press, New York and London.Google Scholar
Simon, B. 1979b. Maximal and minimal Schrödinger forms. J. Operator Theory, 1(1), 37–47.Google Scholar
Simon, B. 1980. The classical limit of quantum partition functions. Comm. Math. Phys., 71(3), 247–276.Google Scholar
Simon, B. 1982. Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.), 7(3), 447–526.Google Scholar
Simon, B. 1983. Nonclassical eigenvalue asymptotics. J. Funct. Anal., 53(1), 84–98.Google Scholar
Simon, B. 1992. The Neumann Laplacian of a jelly roll. Proc. Amer. Math. Soc., 114(3), 783–785.Google Scholar
Simon, B. 2005. Trace Ideals and their Applications. Second edn. Mathematical Surveys and Monographs, vol. 120. Amer. Math. Soc., Providence, RI.Google Scholar
Simon, B. 2011. Convexity: An Analytic Viewpoint. Cambridge Tracts in Mathematics, vol. 187. Cambridge University Press, Cambridge.Google Scholar
Simon, B. 2015a. A Comprehensive Course in Analysis: Part 1, Real Analysis. Amer. Math. Soc., Providence, RI.Google Scholar
Simon, B. 2015b. A Comprehensive Course in Analysis: Part 3, Harmonic Analysis. Amer. Math. Soc., Providence, RI.Google Scholar
Simon, B. 2015c. A Comprehensive Course in Analysis: Part 4, Operator Theory. Amer. Math. Soc., Providence, RI.Google Scholar
Smirnov, V. I. 1964. A Course of Higher Mathematics. Vol. V [Integration and Functional Analysis]. Translated by Brown, D. E.; translation edited by Sneddon, I.N.. Pergamon Press, Oxford and New York; Addison-Wesley Publishing Co., Inc., Reading, MA and London.Google Scholar
Sobolev, A. V. 1993. The Efimov effect. Discrete spectrum asymptotics. Comm. Math. Phys., 156(1), 101–126.Google Scholar
Sobolev, A. V. 1996. On the Lieb–Thirring estimates for the Pauli operator. Duke Math. J., 82(3), 607–635.Google Scholar
Sobolev, S. L. 1935. Le problème de Cauchy dans l’espace des fonctionelles. Dokl. Akad. Nauk SSSR, 3(7), 291–294.Google Scholar
Sobolev, S. L. 1936. On some estimates relating to families of functions having derivatives that are square integrable. Dokl. Akad. Nauk SSSR, 1, 267–270.Google Scholar
Soboleff, S. 1938. Sur un théorème d’analyse fonctionnelle. Rec. Math. Moscou, n. Ser., 4, 471–497.Google Scholar
Sobolev, S. L. 1963. Some Applications of Functional Analysis in Mathematical Physics. Translated from the third Russian edition by McFaden, H. H., with comments by Palamodov, V. P., 1963. Translations of Mathematical Monographs, vol. 90. Amer. Math. Soc., Providence, RI.Google Scholar
Solomyak, M. 1994. Piecewise-polynomial approximation of functions from Hl((0, 1)d), 2l = d, and applications to the spectral theory of the Schrödinger operator. Israel J. Math., 86(1–3), 253–275.Google Scholar
Solomyak, M. 1998. On the discrete spectrum of a class of problems involving the Neumann Laplacian in unbounded domains. Pages 233–251 of: Voronezh Winter Mathematical Schools. Kuchment, P. and Lin, V. (editors). Amer. Math. Soc. Transl. Ser. 2, vol. 184. Amer. Math. Soc., Providence, RI.Google Scholar
Sommerfeld, A. 1910. Die Greensche Funktion der Schwingungsgleichung für ein beliebiges Gebiet. Phys. Z., 11, 1057–1066.Google Scholar
Stein, E. M. 1970. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ.Google Scholar
Stein, E. M., and Shakarchi, R. 2005. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis, vol. 3. Princeton University Press, Princeton, NJ.Google Scholar
Stein, E. M., and Weiss, G. 1971. Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J.Google Scholar
Steklov, V. A. 1896-97. On the differential equations of mathematical physics. Mat. Sb., 19(1), 469–585.Google Scholar
Stepin, S. A. 2001. The Birman–Schwinger principle and the Nelkin conjecture in neutron transport theory. Dokl. Akad. Nauk, 380(1), 19–22. English translation in Dokl. Math., 64(2), 152–155. (2001).Google Scholar
Strichartz, R. S. 1996. Estimates for sums of eigenvalues for domains in homogeneous spaces. J. Funct. Anal., 137(1), 152–190.Google Scholar
Stubbe, J. 2010. Universal monotonicity of eigenvalue moments and sharp Lieb–Thirring inequalities. J. Eur. Math. Soc. (JEMS), 12(6), 1347–1353.Google Scholar
Stummel, F. 1956. Singuläre elliptische Differentialoperatoren in Hilbertschen Räumen. Math. Ann., 132, 150–176.Google Scholar
Sunada, T. 1985. Riemannian coverings and isospectral manifolds. Ann. of Math. (2), 121(1), 169–186.Google Scholar
Sz. Nagy, B. 1941. Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung. Acta Sci. Math., 10, 64–74.Google Scholar
Szegö, G. 1954. Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal., 3, 343–356.Google Scholar
Talenti, G. 1969. Osservazioni sopra una classe di disuguaglianze. Rend. Sem. Mat. Fis. Milano, 39, 171–185.Google Scholar
Talenti, G. 1976. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4), 110, 353–372.Google Scholar
Tamura, H. 1974a. The asymptotic distribution of the lower part eigenvalues for elliptic operators. Proc. Japan Acad., 50, 185–187.Google Scholar
Tamura, H. 1974b. The asymptotic eigenvalue distribution for non-smooth elliptic operators. Proc. Japan Acad., 50, 19–22.Google Scholar
Taylor, M. E. 1981. Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton, NJ.Google Scholar
Temam, R. 1997. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Second edn. Applied Mathematical Sciences, vol. 68. Springer-Verlag, New York.Google Scholar
Teschl, G. 2014. Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators. Second edn. Graduate Studies in Mathematics, vol. 157. Amer. Math. Soc., Providence, RI.Google Scholar
Titchmarsh, E. C. 1958. Eigenfunction Expansions Associated with Second-order Differential Equations. Vol. 2. Clarendon Press, Oxford.Google Scholar
Tomaselli, G. 1969. A class of inequalities. Boll. Un. Mat. Ital. (4), 2, 622–631.Google Scholar
Urakawa, H. 1984. Lower bounds for the eigenvalues of the fixed vibrating membrane problems. Tohoku Mathematical Journal, 36(2), 185–189.Google Scholar
van den Berg, M. 1984. On the spectrum of the Dirichlet Laplacian for horn-shaped regions in Rn with infinite volume. J. Funct. Anal., 58(2), 150–156.Google Scholar
van den Berg, M. 1992. On the spectral counting function for the Dirichlet Laplacian. J. Funct. Anal., 107(2), 352–361.Google Scholar
van den Berg, M., and Lianantonakis, M. 2001. Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions. Indiana Univ. Math. J., 50(1), 299–333.Google Scholar
Vasil’ev, D. G. 1984. Two-term asymptotic behavior of the spectrum of a boundary value problem in interior reflection of general form. Funktsional. Anal. i Prilozhen., 18(4), 1–13, 96. English translation in Functional Anal. Appl., 18, 267–277, (1984).Google Scholar
Vasil’ev, D. G. 1986. Two-term asymptotic behavior of the spectrum of a boundary value problem in the case of a piecewise smooth boundary. Dokl. Akad. Nauk SSSR, 286(5), 1043–1046. English translation in Soviet Math. Dokl., 33(1), 227–230, (1986).Google Scholar
von Neumann, J. 1930. Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann., 102(1), 49–131.Google Scholar
Vulis, I. L., and Solomjak, M. Z. 1974. Spectral asymptotic analysis for degenerate second order elliptic operators. Izv. Akad. Nauk SSSR Ser. Mat., 38, 1362–1392. English translation in Math. USSR Izv., 8(6), 1343–1371, (1974).Google Scholar
Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. Second edn. Reprinted 1995 in Cambridge Mathematical Library. Cambridge University Press, Cambridge.Google Scholar
Weidl, T. 1996. On the Lieb–Thirring constants Lγ,1 for γ ≥ 1/2. Comm. Math. Phys., 178(1), 135–146.Google Scholar
Weidl, T. 1999a. Another look at Cwikel’s inequality. Pages 247–254 of: Differential Operators and Spectral Theory. Amer. Math. Soc. Transl. Ser. 2, vol. 189. Providence, RI: Amer. Math. Soc.Google Scholar
Weidl, T. 1999b. Remarks on virtual bound states for semi-bounded operators. Comm. Partial Diff Equations, 24(1–2), 25–60.Google Scholar
Weidl, T. 2008. Improved Berezin–Li–Yau inequalities with a remainder term. Pages 253–263 of: Spectral Theory of Differential Operators. Amer. Math. Soc. Transl. Ser. 2, vol. 225. Amer. Math. Soc., Providence, RI.Google Scholar
Weinberger, H. F. 1956. An isoperimetric inequality for the N -dimensional free membrane problem. J. Rational Mech. Anal., 5, 633–636.Google Scholar
Weinstein, M. I. 1983. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87(4), 567–576.Google Scholar
Weyl, H. 1909. Über beschränkte quadratische Formen, deren Differenz vollstetig ist. Palermo Rend., 27(1), 373–392.Google Scholar
Weyl, H. 1911. Über die asymptotische Verteilung der Eigenwerte. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 1911, 110–117.Google Scholar
Weyl, H. 1912a. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann., 71(4), 441–479.Google Scholar
Weyl, H. 1912b. Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung. J. Reine Angew. Math., 141, 1–11.Google Scholar
Weyl, H. 1912c. Über das Spektrum der Hohlraumstrahlung. J. Reine Angew. Math., 141, 163–181.Google Scholar
Weyl, H. 1913. Über die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetze. J. Reine Angew. Math., 143, 177–202.Google Scholar
Yafaev, D. R. 1982. The low energy scattering for slowly decreasing potentials. Comm. Math. Phys., 85(2), 177–196.Google Scholar
Yafaev, D. R. 1992. Mathematical Scattering Theory: General Theory. Translated from the Russian by Schulenberger, J. R.. Translations of Mathematical Monographs, vol. 105. Amer. Math. Soc., Providence, RI.Google Scholar
Yafaev, D. R. 2010. Mathematical Scattering Theory: Analytic Theory. Mathematical Surveys and Monographs, vol. 158. Amer. Math. Soc., Providence, RI.Google Scholar
Zaharov, V. E., and Faddeev, L. D. 1971. The Korteweg–de Vries equation is a fully integrable Hamiltonian system. Funkcional. Anal. i Priložen., 5(4), 18–27. English translation in Funct. Anal. Appl., 5, 280–287, (1972).Google Scholar
Zelditch, S. 2004. Inverse spectral problem for analytic domains. I. Balian–Bloch trace formula. Comm. Math. Phys., 248(2), 357–407.Google Scholar
Zelditch, S. 2009. Inverse spectral problem for analytic domains. II. ℤ2-symmetric domains. Ann. of Math. (2), 170(1), 205–269.Google Scholar
Zworski, M. 2012. Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. Amer. Math. Soc., Providence, RI.Google Scholar

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  • References
  • Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
  • Book: Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities
  • Online publication: 03 November 2022
  • Chapter DOI: https://doi.org/10.1017/9781009218436.014
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  • References
  • Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
  • Book: Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities
  • Online publication: 03 November 2022
  • Chapter DOI: https://doi.org/10.1017/9781009218436.014
Available formats
×

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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
  • Book: Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities
  • Online publication: 03 November 2022
  • Chapter DOI: https://doi.org/10.1017/9781009218436.014
Available formats
×