Singular perturbations, regularization and extension theory

Abstract

For nonpositive singular potentials in quantum mechanics it can happen that the Schrödinger operator is not essentially self-adjoint on a natural domain of definition or not semibounded from below. In this case we have a lot of self-adjoint extensions each of them is a candidate for the right physical Hamiltonian of the system. Hence the problem arises to single out the right physical self-adjoint extension. Usually this problem is solved as follows. At first one has to approximate the singular potential by a sequence of bounded potentials (cut-off approximation). After that one has to show that the arising sequence of Schrödinger operators converges in the strong resolvent sense to one of the self-adjoint extensions if the cut-off approximation tends to the singular potential. The so determined self-adjoint extensions is regarded as the right physical Hamiltoninan. Very often the right physical Hamiltonian coincides with the Friedrichs extension.