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Schrödinger Operators on a Half-Line with Inverse SquarePotentials

Published online by Cambridge University Press:  17 July 2014

H. Kovařík  and
F. Truc
H. Kovařík*
Affiliation:
DICATAM, Sezione di Matematica, Università di Brescia, Via Branze, 38 - 25123 Brescia, Italy
F. Truc
Affiliation:
Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d’Hères Cedex, France
*
⋆Corresponding author. E-mail: hynek.kovarik@unibs.it
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Abstract

We consider Schrödinger operators Hα given by equation(1.1) below. We study the asymptotic behavior of the spectral density E(Hα,λ)for λ → 0 andthe L1 →L∞ dispersive estimates associated to theevolution operator e−itHα.In particular we prove that for positive values of α, the spectral densityE(Hα,λ)tends to zero as λ →0 with higher speed compared to the spectral density of Schrödingeroperators with a short-range potential V. We then show how the long time behavior ofe−itHαdepends on α.More precisely we show that the decay rate of e−itHαfor t → ∞ canbe made arbitrarily large provided we choose α large enough and consider a suitable operatornorm.

Keywords

inverse square potentialsdispersive estimates
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 170 - 176
Copyright
© EDP Sciences, 2014

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References

M. Abramowitz, I. Stegun. Handbook of mathematical functions. National Bureau of Standards, 1964.
Burq, N., Planchon, F., Stalker, J., Tahvildar-Zadeh, S.. Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. J. Funct. Anal., 203 (2003), 519–549. CrossRefGoogle Scholar
Burq, N., Planchon, F., Stalker, J., Tahvildar-Zadeh, S.. Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J., 53 (2004), 1665–1680. CrossRefGoogle Scholar
N. Dunford, J.T. Schwartz. Linear Operators, Part II. New York, 1988.
A. Erdelyi: Tables of integral transforms, Vol. 2. McGraw-Hill New York, 1954.
Erdogan, M. B., Green, W. R.. A weighted dispersive estimate for Schrödinger operators in dimension two. Comm. Math. Phys., 319 (2013), 791–811. CrossRefGoogle Scholar
Fanelli, L., Felli, V., Fontelos, M. A., Primo, A.. Time decay of scaling critical electromagnetic Schrödinger flows. Comm. Math. Phys., 324 (2013), 1033–1067. CrossRefGoogle Scholar
Goldberg, M.. Transport in the one-dimensional Schrödinger equation. Proc. Amer. Math. Soc., 135 (2007), 3171–3179. CrossRefGoogle Scholar
Goldberg, M., Schlag, W.. Dispersive estimates for Schrödinger operators in dimensions one and three. Comm. Math. Phys., 251 (2004), 157–178. CrossRefGoogle Scholar
M. Goldberg, L. Vega, N Visciglia. Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials. Int. Math Res. Not, (2006), article ID 13927.
G. Grillo, H. Kovařík. Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov-Bohm magnetic field. J. Differential Equations, 256 (2014), 3889–3911.
Jensen, A., Kato, T.. Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J., 46 (1979), 583–611. CrossRefGoogle Scholar
Jensen, A., Nenciu, G.. Schrödinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds. Proc. Indian Acad. Sci. Math. Sci., 116 (2006), 375–392. CrossRefGoogle Scholar
Kovařík, H.. Heat kernels of two-dimensional magnetic Schrödinger and Pauli operators. Calc. Var. Partial Differential Equations, 44 (2012), 351–374. CrossRefGoogle Scholar
Milman, P.D., Semenov, Yu. A.. Heat kernel bounds and desingularizing weights. J. Funct. Anal., 202 (2003), 1–24. CrossRefGoogle Scholar
Milman, P.D., Semenov, Yu. A.. Global heat kernel bounds via desingularizing weights. J. Funct. Anal., 212 (2004), 373–398. CrossRefGoogle Scholar
Murata, M.. Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal., 49 (1982), 10–56. CrossRefGoogle Scholar
Schlag, W.. Dispersive estimates for Schrödinger operators in dimension two. Comm. Math. Phys., 257 (2005), 87–117. CrossRefGoogle Scholar
W. Schlag. Dispersive estimates for Schrödinger operators: a survey. Mathematical aspects of nonlinear dispersive equations. 255-285, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.
G. Teschl. Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators. American Mathematical Society Providence, Rhode Island, 2009.
Wang, X.P.. Asymptotic expansion in time of the Schrödingier group on conical manifolds. Ann. Inst. Fourier, 56 (2006), 1903–1945. CrossRefGoogle Scholar
Weder, R.. Lp-Lp’ Estimates for the Schrödinger Equation on the Line and Inverse Scattering for the Nonlinear Schrödinger Equation with a Potential. J. Funct. Anal., 170 (2000), 37–68. CrossRefGoogle Scholar
Weder, R.. The Lp-Lp estimate for the Schrödinger equation on the half-line. J. Math. Anal. Appl., 281 (2003), 233–243. CrossRefGoogle Scholar