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The existence of eigenvalues of infinite multiplicity for the Schrödinger operator

Published online by Cambridge University Press:  14 November 2011

Christine R. Thurlow  and
Michael S.P. Eastham
Christine R. Thurlow
Affiliation:
St. Hilda's College, Oxford†
Michael S.P. Eastham
Affiliation:
Chelsea College, University of London
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Synopsis

It is shown that eigenvalues of infinite multiplicity can exist for the Schrödinger equation holding in the whole N-dimensional space RN(N ≧ 2). In the example which is constructed, the potential is separable and bounded in RN, and the method is an application of inverse spectral theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 1-2 , 1980 , pp. 61 - 64
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Berezanskii, Ju. M.. Expansions in eigenfunctions of self-adjoint operators (Providence, R.I.A.M.S., 1968).CrossRefGoogle Scholar
2Blokh, A. Sh.. On the determination of a differential equation by its spectral matrix function Doki. Akad. Nauk SSSR 92 (1953), 209–212.Google Scholar
3Eastham, M. S. P. and McLeod, J. B.. The existence of eigenvalues embedded in the continuos spectrum of ordinary differential operators. Proc. Roy. Soc. Edinburgh Sect. A 79 (1977), 25–34.CrossRefGoogle Scholar
4Glazman, I. M.. Direct methods of qualitative spectral analysis of singular differential operater (Jerusalem: I.P.S.T., 1965).Google Scholar
5Kato, T.. Growth properties of solutions of the reduced wave equation with a variable coefficien Comm. Pure Appl. Math. 12 (1959), 403–425.CrossRefGoogle Scholar
6Kato, T.. Schrödinger operators with singular potentials. Israel J. Math. 13 (1972), 135–148.CrossRefGoogle Scholar
7Levitán, B. M. and Gasymov, M. G.. Determination of a differential equation by two of its specter Russian Math. Surveys 19 (1964), 1–63.CrossRefGoogle Scholar
8Naimark, M. A.. Linear differential operators, pt 2 (London: Harrap, 1968).Google Scholar
9Putnam, C. R.. A sufficient condition for an infinite discrete spectrum. Quart. Appl. Math. 17 (1953), 484–487.Google Scholar
10Schechter, M.. Discreteness of the singular spectrum for Schrödinger operators. Math. Proc. Cam. Philos. Soc. 80 (1976), 121–133.CrossRefGoogle Scholar
11Thomas, L. E.. Time dependent approach to scattering from impurities in a crystal. Comm. Mai Phys. 33 (1974), 335–343.CrossRefGoogle Scholar
12Thurlow, C. R.. A generalisation of the inverse spectral theorem of Levitán and Gasymov. Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 185–196.CrossRefGoogle Scholar
13Thurlow, C. R.. The point-continuous spectrum for ordinary second-order differential equation Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 197–211.CrossRefGoogle Scholar
14Titchmarsh, E. C.. Eigenfunction expansions, 2nd edn, pt 1 (Oxford Univ. Press, 1962).Google Scholar