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Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions

Published online by Cambridge University Press:  20 November 2018

Paul Binding  and
Branko Ćurgus
Paul Binding
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, email: binding@ucalgary.ca
Branko Ćurgus
Affiliation:
Department of Mathematics, Western Washington University, Bellingham, WA 98225 USA, email: curgus@cc.wwu.edu
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Abstract

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Form domains are characterized for regular $2n$-th order differential equations subject to general self-adjoint boundary conditions depending affinely on the eigenparameter. Corresponding modes of convergence for eigenfunction expansions are studied, including uniform convergence of the first $n\,-\,1$ derivatives.

Keywords

47E0534B0947B5047B2534L10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 6 , 01 December 2002 , pp. 1142 - 1164
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Azizov, T. Ya. and Iokhvidov, I. S., Linear operators in spaces with an indefinite metric. John Wiley & Sons, 1990.Google Scholar
[2] Binding, P. A., A canonical form for self-adjoint pencils in Hilbert space. Integral Equations Operator Theory 12 (1984), 324–342.Google Scholar
[3] Binding, P. A. and Browne, P. J., Eigencurves for two parameter self-adjoint ordinary differential equations of even order. J. Differential Equations 79 (1989), 289–303.Google Scholar
[4] Binding, P. A. and Browne, P. J., Form domains for sectorial operators related to generalized Sturm-Liouville problems. Quart. J. Math. Oxford Ser. (2) 50 (1999), 155–178.Google Scholar
[5] Bognàr, J., Indefinite inner product spaces. Springer-Verlag, 1974.Google Scholar
[6] Coddington, E. A. and de Snoo, H. S. V., Regular boundary value problems associated with pairs of ordinary differential expressions. Lecture Notes in Math. 858, Springer-Verlag, 1981.Google Scholar
[7] Ćurgus, B., On the regularity of the critical point infinity of definitizable operators. Integral Equations Operator Theory (4) 8 (1985), 462–488.Google Scholar
[8] Ćurgus, B., Definitizable extensions of positive symmetric operators in a Krein space. Integral Equations Operator Theory 12 (1989), 615–631.Google Scholar
[9] Ćurgus, B., Dijksma, A. and Read, T., The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces. Linear Algebra Appl. 329 (2001), 97–136.Google Scholar
[10] Ćurgus, B. and Langer, H., A Krein space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differential Equations 79 (1989), 31–61.Google Scholar
[11] Ćurgus, B. and Najman, B., Quasi-uniformly positive operators in Krein space. In: Operator theory and boundary eigenvalue problems (Vienna, 1993), Oper. Theory Adv. Appl. 80, Birkhäuser, 1995, 90–99.Google Scholar
[12] Dijksma, A., Eigenfunction expansions for a class of J-self-adjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 1–27.Google Scholar
[13] Dijksma, A. and Langer, H., Operator theory and ordinary differential operators. Fields Inst. Monogr. 3, Amer.Math. Soc., Providence, RI, 1996, 73–139.Google Scholar
[14] Fulton, C. T., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 293–308.Google Scholar
[15] Hinton, D. B., An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition. Quart. J. Math. Oxford Ser. (2) 30 (1979), 33–42.Google Scholar
[16] Kato, T., Perturbation Theory of Linear Operators. Springer-Verlag, 1966.Google Scholar
[17] Kemp, R. R. D., Operators on L2(I) ⊕ Cr . Canad. J. Math. 39 (1987), 33–53.Google Scholar
[18] Krein, M. G., Theory of self-adjoint extensions of semibounded Hermitian transformations and its applications II. Mat. Sb. 21(63)(1947), 365–404.Google Scholar
[19] Langer, H., Spectral function of definitizable operators in Krein spaces. Functional Analysis, Proceedings, Dubrovnik 1981. Lecture Notes in Mathematics 948, Springer-Verlag, 1982, 1–46.Google Scholar
[20] Naimark, M. A., Linear differential operators. Part II: Linear differential operators in Hilbert space. Frederick Ungar Publishing Co., 1968.Google Scholar
[21] Shkalikov, A. A., Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. English transl.: J. Soviet Math. (6) 33 (1986), 1311–1342.Google Scholar
[22] Tretter, C., Nonself-adjoint spectral problems for linear pencils N − λP of ordinary differential operators with λ-linear boundary conditions: completeness results. Integral Equations Operator Theory 26 (1996), 222–248.Google Scholar
[23] Walter, J., Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Math. Z. 133 (1973), 301–312.Google Scholar
[24] Weidmann, J., Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics 1258, Springer-Verlag, 1987.Google Scholar