On the Birman problem in the theory of nonnegative symmetric operators with compact inverse

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.