Locality of quadratic forms for point perturbations of Schrödinger operators. (English. Russian original) Zbl 1058.47007

Pankrashkin, K. V.

Math. Notes 70, No. 3, 384-391 (2001); translation from Mat. Zametki 70, No. 3, 425-433 (2001).

The point perturbation \(H\) of the self-adjoint operator \(H^0\) in \(L^2 (\mathbb{R}^\nu)\) has the form \(H=H^0+\sum_{a\in{A}}\lambda^a \delta(\cdot-a)\), where \(A\) is a discrete subset of \(\mathbb{R}^\nu\) and \(\delta\) is the Dirac delta-function. Similar Hamiltonians with regular periodic perturbations (rather than \(\delta\)-functions) usually have spectra with “reasonable” regular structure, while the spectral structure of corresponding Hamiltonians with point perturbations can be more exotic. In particular, a Hamiltonian with point perturbation can have a spectrum with fractal regions. One way to preserve the regular structure of the spectrum after the replacement of a regular potential by the point interaction is to use the point perturbations obeying the condition of form-locality (i.e., locality in the sense of forms) [V. A. Geiler and K. V. Pankrashkin, Oper. Theory Adv. Appl. 108, 259–265 (1999; Zbl 0972.81037)].
The paper under review deals with the problem of finding, in the framework of Kreĭn’s theory of self-adjoint extensions, the point perturbations of a Schrödinger operator which are form-local. Some necessary and sufficient condition for the locality in the sense of forms of point perturbations of Schrödinger operators is found in the cases of two and three dimensions.

Reviewer: Pavel Dyshlovenko (Ul’yanovsk)

MSC:

47A55	Perturbation theory of linear operators
47A10	Spectrum, resolvent
47N50	Applications of operator theory in the physical sciences
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15	Perturbation theories for operators and differential equations in quantum theory

Keywords:

Krein theory; self-adjoint extension; singular perturbation; point interaction; semi-bounded operator