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Non-existence of eigenvalues of Schrödinger operators

Published online by Cambridge University Press:  14 February 2012

H. Kalf
H. Kalf
Affiliation:
Institut für Mathematik, RWTH Aachen
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Synopsis

The paper provides conditions which enstlre that the Schrödinger operator

defined on an exterior domain has no eigenvalues on a certain half-ray. These conditions are in terms of weak local assumptions on

The proof uses Kato's ideas [16] in conjunction with the physicists' “commutator proof” of the quantum mechanical virial theorem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 1-2 , 1977 , pp. 157 - 172
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Agmon, S.. Lower bounds for solutions of Schrödinger-type equations in unbounded domains. Proc. Internal Conf. Functional Analysis Related Topics, Tokyo (1969), 216–224.Google Scholar
2Agmon, S.. Lower bounds for solutions of Schrödinger equations. J. Analyse Math. 23 (1970), 1–25.CrossRefGoogle Scholar
3Albeverio, S.. On bound states in the continuum of N-body systems and the virial theorem. Ann. Physics 71 (1972), 167–276.CrossRefGoogle Scholar
4Aronszajn, N.. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J.Math. Pures Appl. 36 (1957), 235–249.Google Scholar
5Avron, J. E. and Herbst, I. W.. Spectral and scattering theory of Schrödinger operators related to the Stark effect. Princeton Univ. Preprint, 1976.Google Scholar
6Balslev, E.. Absence of positive eigenvalues of Schrodinger operators. Arch. Rational Mech. Anal. 59 (1975), 343–357.CrossRefGoogle Scholar
7Cordes, H. O.. Über die eindeutige Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1956), 239–258.Google Scholar
8Hellwig, G.. Differential operators of mathematical physics (Reading: Addison, 1967).Google Scholar
9Ikebe, T. and Kato, T.. Uniqueness of the self-adjoint extensions of singular elliptic differential operators. Arch. Rational Mech. Anal. 9 (1962), 77–92.CrossRefGoogle Scholar
10Ikebe, T. and Uchiyama, J.. On the asymptotic behaviour of eigenfunctions of second-order elliptic operators. J.Math. Kyoto Univ. 11 (1971), 425–448.Google Scholar
11Jager, W.. Über das Dirichletsche AuBenraumproblem fur die Schwingungsgleichune. Math. Z. 95 (1967), 299–323.CrossRefGoogle Scholar
12Jansen, K.-H. and Kalf, H.. On positive eigenvalues of one-body Schrödinger operators: remarks on papers by.Agmon and Simon. Comm. Pure Appl. Math. 28 (1975), 747–;752.CrossRefGoogle Scholar
13Jörgens, K.. Über das wesentliche Spektrum elliptischer Differentialoperatoren vom Schrödinger-Typ (Heidelberg: Univ. Technicher Bericht, 1965).Google Scholar
14Jörgens, K.. Zur Spektraltheorie der Schrodinger-Operatoren. Math. Z. 96(1967), 355–372.CrossRefGoogle Scholar
15Kalf, H.. The quantum mechanical virial theorem and the absence of positive energy bound states of Schrodinger operators. Israel J. Math. 20 (1975), 57–69.CrossRefGoogle Scholar
16Kato, T.. Growth properties of solutions of the reduced wave equation with a variable coefficient. Comm. Pure Appl Math. 12 (1959), 403–425.CrossRefGoogle Scholar
17Ladyzhenskaya, O. A. and Ural'tseva, N. N.. Linear and quasilinear elliptic equations (London:, Academic Press, 1968).Google Scholar
18Landau, L. D. and Lifshitz, E. M.. jQuantum mechanics (London: Pergamon, 1959).Google Scholar
19Neumann, J. v. and Wigner, E.. Über merkwürdige diskrete Eigenwerte. Phys. Z. 30 (1929), 465–467.Google Scholar
20Odeh, F.. Note on differential operators with a purely continuous spectrum. Proc. Amer. Math. Soc. 16 (1965), 363–366.CrossRefGoogle Scholar
21Oseen, C. W.. Deux remarques sur la méthode des perturbations dans la mécanique ondulatoire. Ark. Mat. A 25 (1937), no. 2.Google Scholar
22Protter, M. H.. Unique continuation for elliptic equations. Trans. Amer. Math. Soc. 95 (1960), 81–91.CrossRefGoogle Scholar
23Rohde, H.-W.. Ein Kriterium fur das Fehlen von Eigenwerten elliptischer Differentialoperatoren. Math. Z. 112 (1969), 375–388.CrossRefGoogle Scholar
24Schechter, M.. Spectra of partial differential operators (Amsterdam: North Holland, 1971).Google Scholar
25Simon, B.. On positive eigenvalues for one-body Schrödinger Operators. Comm.Pure Appl. Math. 22 (1969), 531–538.CrossRefGoogle Scholar
25Simon, B.. Absence of positive eigenvalues in a class of multi-particle quantum systems. Math. Ann. 207 (1974), 133–138.CrossRefGoogle Scholar
27Uchiyama, J.. Lower bounds of growth order of solutions of Schrödinger equations with homogeneous potentials. Publ. Res. Inst. Math. Sci. 10 (1975), 425–444.CrossRefGoogle Scholar
28Weidmann, J.. The virial theorem and its application to the spectral theory of Schrödinger operators. Bull. Amer. Math. Soc. 73 (1967), 452–456.CrossRefGoogle Scholar