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

Large classes of nonnegative Schrödinger operators on <span class="mathjax-tex">\(\Bbb R^2\)</span> and <span class="mathjax-tex">\(\Bbb R^3\)</span> with the following properties are described: </p><p> 1. The restriction of each of these operators to an appropriate unbounded set of measure zero in <span class="mathjax-tex">\(\Bbb R^2\)</span> (in <span class="mathjax-tex">\(\Bbb R^3\)</span>) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent; </p><p> 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. </p><p> The obtained results give a solution of a problem by M. S. Birman.