Locality of quadratic forms for point perturbations of Schrödinger operators. (Q1809990)

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) [\textit{V. A. Geiler} and \textit{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.