Weyl function of a Hermitian operator and its connection with characteristic function

Abstract

Let $A$ be a densely defined symmetric operator with equal deficiency indices in a Hilbert space. We introduce the notion of a Weyl function $M(z)$ of $A$ corresponding to an ordinary boundary triplet of the operator $A^*$ and then investigate its basic properties. In particular, a connection with Krein-Langer Q-functions and Krein's type formula for resolvents is discovered. Using this new connection, we show that the resolvent comparability of two proper extensions is equivalent to that of the corresponding boundary operators. Moreover, we show that the number of negative eigenvalues of a self-adjoint extension $A_B=A_B^*$ of a non-negative operator $A$ equals the number of negative eigenvalues of $B-M(0-)$, where $B$ is the boundary operator of $A_B$ and $M(0-)$ is the left limit of the Weyl function at zero. Also, we introduce the class of almost solvable extensions of $A$. A characteristic function (in the sense of A. V. Shtraus) of an almost solvable extension is expressed by means of the Weyl function and the corresponding boundary operator. Analytic properties of characteristic functions are completely characterized. The main results are applied to ordinary differential operators, Sturm-Liouville operators with unbounded operator potentials, Shr\"odinger operators and Laplacians on domains with a non-smooth boundary. These results were substantially elaborated and published later in the following papers: 1. V.A. Derkach and M.M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), 1-95. 2. --- Characteristic functions of almost solvable extensions of a Hermitian operators, Ukr. Mat. Zh. 44 (1992), 435-459. 3. --- The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), 141-242.