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Elliptic multiparameter eigenvalue problems

Published online by Cambridge University Press:  20 January 2009

P. A. Binding and
K. Seddighi
P. A. Binding
Affiliation:
Department of Mathematics snd Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4
K. Seddighi
Affiliation:
Department of Mathematics, University of Shiraz, Iran
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We study the eigenproblem

where

and Tm, Vmn are self-adjoint operators on separable Hilbert spaces Hm. We assume the Tm to be bounded below with compact resolvents, and the Vmn to be bounded and to satisfy an “ellipticity” condition. If k = 1 then ellipticity is automatic, and if each Tm is positive definite then the problem is “left definite”.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 2 , June 1987, pp. 215 - 228
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1Arscott, F. M., Periodic Differential Equations: An Introduction to Mathieu, L´me and Allied Functions (Macmillan Press, 1964).Google Scholar
2Atkinson, F. V., Multiparameter Eigenvalue Problems, Vol. 1 (Academic Press, 1972).Google Scholar
3Binding, P. A., Multiparameter definiteness conditions, Proc. Roy. Soc. Edinburgh 89A (1981), 319–332.CrossRefGoogle Scholar
4Binding, P. A., Left definite multiparameter eigenvalue problems, Trans. Amer. Math. Soc. 272 (1982), 475–486.CrossRefGoogle Scholar
5Binding, P. A. and Browne, P. J., A variational approach to multiparameter eigenvalue problems in Hilbert space, SIAM J. Math. Anal. 9 (1978), 1054–1067.CrossRefGoogle Scholar
6Binding, P. A. and Seddighi, K., On root vectors of self-adjoint pencils, J. Funct. Anal., to appear.Google Scholar
7Bognar, J., Indefinite Inner Product Spaces (Springer-Verlag, 1974).CrossRefGoogle Scholar
8Daho, K. and Langer, H., Sturm-Liouville operators with an indefinite weight function, Proc. Roy. Soc. Edinburgh 78A (1977), 161–191.CrossRefGoogle Scholar
9Faierman, M., Expansions in eigenfunctions of a two-parameter system of differential equations I-IV, to appear.Google Scholar
10Fleckinger, J. and Mingarelli, A. B., On the eigenvalues of non-definite elliptic operators, Differential Equations (ed. Knowles, I. W., Lewis, R. T.) (North-Holland, 1984), 219–227.Google Scholar
11Gantmacher, F. R., The Theory of Matrices, Vol. 2 (Chelsea, 1959).Google Scholar
12Gohberg, I., Lancaster, P. and Rodman, L., Matrices and Indefinite Scalar Products (Birkhäuser-Verlag, 1983).Google Scholar
13Greenlee, W. M., Double unconditional bases associated with a quadratic characteristic parameter problem, J. Funct. Anal. 15 (1974), 306–339.CrossRefGoogle Scholar
14Hilbert, D., Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen, 1912 (reprinted by Chelsea, 1953).Google Scholar
15Kallstrom, A. and Sleeman, B. D., A left definite multiparameter eigenvalue problem in ordinary differential equations, Proc. Roy. Soc. Edinburgh 74A (1976), 145–155.CrossRefGoogle Scholar
16Kato, T., Perturbation Theory for Linear Operators (Springer-Verlag, 1966).Google Scholar
17Mal-Cev, A. I., Foundations of Linear Algebra (Freeman, 1963).Google Scholar
18Roach, G. F. and Sleeman, B. D., On the spectral theory of operator bundles II, Applicable Anal. 9 (1979), 29–36.CrossRefGoogle Scholar
19Sleeman, B. D., Multiparameter Spectral Theory in Hilbert Space (Pitman, 1978).CrossRefGoogle Scholar
20Volkmer, H., On multiparameter theory, J. Math. Anal. Appl. 86 (1982), 44–53.CrossRefGoogle Scholar
21Volkmer, H., On the completeness of eigenvectors of right definite multiparameter problems, Proc. Roy. Soc. Edinburgh 96A (1984), 69–78.CrossRefGoogle Scholar