Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-30T13:00:20.145Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the domain characterization of certain Schrödinger operators with strongly singular potentials

Published online by Cambridge University Press:  14 November 2011

Hubert Kalf
Hubert Kalf
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darmstadt, B.R.D.
Get access Rights & Permissions [Opens in a new window]

Synopsis

For β > β0: = 1 −[(n − 2)/2]2 and n ≧ 2, it was recently shown by Simon that the self-adjoint operator associated with −Δ + βr−2 in L2(ℝn) has domain H2(ℝn) ∩D(r−2) the constant β0 being the best possible. An alternative proof of this result is given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 125 - 130
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Atkinson, F. V.. On some results of Everitt and Giertz. Proc. Roy. Soc. Edinburgh Sect. A 71 (1972/1973), 151–158.Google Scholar
2Everitt, W. N. and Giertz, M.. Some properties of the domains of certain differential operators. Proc. London Math. Soc. 23 (1971), 301–324.CrossRefGoogle Scholar
3Everitt, W. N. and Giertz, M.. Inequalities and separation for certain partial differential operators. Preprint, Royal Inst. Tech. Stockholm 1973.CrossRefGoogle Scholar
4Everitt, W. N. and Giertz, M.. Inequalities and separation for Schrödinger type operators in L 2(ℝn). Proc. Roy. Soc. Edinburgh Sect. A 79 (1977), 257–265.CrossRefGoogle Scholar
5Faris, W. G.. Self-adjoint operators. Lecture Notes in Mathematics 433 (Berlin: Springer, 1975).Google Scholar
6Glimm, J. and Jaffe, A.. Singular perturbations of self-adjoint operators. Comm. Pure Appl. Math. 22 (1969), 401–414.CrossRefGoogle Scholar
7Kalf, H.. Gauss's theorem and the self-adjointness of Schrödinger operators. Ark. Mat. 18 (1980), 19–47.CrossRefGoogle Scholar
8Kato, T.. Remarks on the selfadjointness and related problems for differential operators. In Spectral Theory of Differential Operators (Knowles, I. W. and Lewis, R. T. eds.), 253–266 (Amsterdam: North-Holland, 1981).Google Scholar
9Reed, M. and Simon, B.. Methods of Modem Mathematical Physics II: Fourier Analysis, Self-Adjointness (New York: Academic Press, 1975).Google Scholar
10Robinson, D. W.. Scattering theory with singular potentials. 1. The two-body problem. Ann. Inst. H. Poincaré Sect. A (N.S.) 21 (1974), 185–215.Google Scholar
11Schmincke, U. W.. Essential selfadjointness of a Schrödinger operator with strongly singular potential. Math. Z. 124 (1972), 47–50.CrossRefGoogle Scholar
12Simon, B.. Hardy and Rellich inequalities in non-integral dimension. J. Operator Theory 9 (1983), 143–146.Google Scholar
13Sohr, H.. Über die Selbstadjungiertheit von Schrödinger-Operatoren. Math. Z. 160 (1978), 255–261.CrossRefGoogle Scholar
14Tafel, W.. Relative Beschränktheit und maximale Differentialoperatoren. Dissertation, Universität München 1979.Google Scholar
15Titchmarsh, E. C.. Eigenfunction Expansions Associated With Second-Order Differential Equations (Oxford: Clarendon Press, 1946).Google Scholar