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On the stability of self-adjointness of Schrödinger operators under positive perturbations

Published online by Cambridge University Press:  14 November 2011

Hans L. Cycon
Hans L. Cycon
Affiliation:
Technische Universität Berlin, Fachbereich Mathematik, B.R.D.
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Synopsis

We introduce a class of essentially self-adjoint Schrödinger operators, where essential self-adjointness is stable under positive potential perturbations. We show that this class is “stable” under certain perturbations and contains operators discussed by Simon and Kato. Finally, an extended essentially self-adjointness criterion is given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 1-2 , 1980 , pp. 165 - 173
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2Cycon, H. L.. Schrödingeroperatoren mit positiven Definitionskernen (Dissertation, TU Berlin, 1979) (D83).Google Scholar
3Ginibre, J.. Auto-adjonction essentielle du Hamiltonien der Schrödinger avec potentiel singulier. Preprint, Laboratoire de Physique Théorique et Haute Energies, 75/22 (1975).Google Scholar
4Goelden, H.-W.. On non-degeneracy of the ground state of Schrödinger operators. Math. Z. 155 (1977), 239–247.CrossRefGoogle Scholar
5Grütter, A.. Wesentliche Selbstadjungiertheit eines Schrödinger-Operators. Math. Z. 135 (1974), 289–291.CrossRefGoogle Scholar
6Jörgens, K.. Wesentliche Selbstadjungiertheit singulärer elliptischer Differentialoperatoren zweiter Ordnung in . Math. Scand. 15 (1964), 5–17.CrossRefGoogle Scholar
7Kalf, H. and Walter, J.. Strongly singular potentials and essential self-adjointness of singular elliptic operators in . J. Functional Analysis 10 (1972), 114–130.CrossRefGoogle Scholar
8Kalf, H. and Walter, J.. Note on a paper of Simon on essentially self-adjoint Schrödinger operators with singular potentials. Arch. Rational Mech. Anal. 52 (1973), 258–260.CrossRefGoogle Scholar
9Kalf, H., Schmincke, U.-W., Walter, J. and Wüst, R.. On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials. Symposium Dundee 1974. Lecture Notes in Mathematics 448, 182–226 (Berlin; Springer, 1975).Google Scholar
10Kato, T.. Perturbation Theory for Linear Operators. (Berlin: Springer, 1966).Google Scholar
11Kato, T.. Schrödinger operators with singular potentials. Israel J. Math. 13 (1972), 135–148.Google Scholar
12Kurss, H.. A limit-point criterion for non-oscillatory Sturm-Liouville differential operators. Proc. Amer. Math. Soc. 18 (1967), 445–449.CrossRefGoogle Scholar
13Levitán, B. M. and Otelbaev, M.. On conditions for self-adjointness of the Schrödinger and Dirac operators. Soviet Math. Dokl. 18 (1977), 1044–1048.Google Scholar
14Reed, M. and Simon, B.. Methods of Modern Mathematical Physics, vol. I, Functional Analysis (New York: Academic Press, 1972).Google Scholar
15Reed, M. and Simon, B.. Methods of Modern Mathematical Physics, vol. II, Fourier Analysis, Self-adjointness (New York: Academic Press, 1975).Google Scholar
16Schechter, M.. Spectra of Partial Differential Operators (Amsterdam: North Holland, 1971).Google Scholar
17Schmincke, U.-W.. Essential self adjointness of a Schrödinger operator with strongly singular potential. Math. Z. 124 (1972), 47–50.CrossRefGoogle Scholar
18Semenov, Yu A.. Schrödinger operators with -potentials. Comm. Math. Phys. 53 (1977), 277–284.CrossRefGoogle Scholar
19Simader, C. G.. Bemerkungen über Schrödinger Operatoren mit stark singulären Potentialen. Math. Z. 138 (1974), 53–70.CrossRefGoogle Scholar
20Simon, B.. Essential self-adjointness of Schrödinger operators with positive potentials. Math. Ann. 201 (1973), 211–220.CrossRefGoogle Scholar
21Simon, B.. Essential self-adjointness of Schrödinger operators with singular potentials. Arch. Rational Mech. Anal. 52 (1973), 44–48.CrossRefGoogle Scholar
22Simon, B.. An abstract Kato's inequality for generators of positivity preserving semigroups. Indiana Univ. Math. J. 26 (1977), 1067–1073.CrossRefGoogle Scholar
23Stetkaer-Hansen, H.. A generalization of a theorem of Wienholtz concerning essential self-adjointness of singular elliptic operators. Math. Scand. 19 (1966), 108–112.Google Scholar
24Walter, J.. Note on a paper by Stetkaer-Hansen concerning essential self-adjointness of Schrödinger operators. Math. Scand. 25 (1969), 94–96.Google Scholar
25Wüst, R.. Generalizations of Rellich's theorem on perturbation of (essentially) self-adjoint operators. Math. Z. 119 (1971), 276–280.Google Scholar
26Wüst, R.. A convergence theorem for selfadjoint operators applicable to Dirac operators with cutoff potentials. Math. Z. 131 (1973), 339–349.CrossRefGoogle Scholar