On the Birman Problem in the Theory of Nonnegative Symmetric Operators with Compact Inverse

Abstract

Large classes of nonnegative Schrödinger operators on \(\Bbb R^2\) and \(\Bbb R^3\) with the following properties are described:

1. The restriction of each of these operators to an appropriate unbounded set of measure zero in \(\Bbb R^2\) (in \(\Bbb R^3\)) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent;

2. Under certain additional assumptions on the potential, the Friedrichs extension of such a restriction has continuous (sometimes absolutely continuous) spectrum filling the positive semiaxis.

The obtained results give a solution of a problem by M. S. Birman.