A general boundary value problem and its Weyl function

Abstract

We study the abstract boundary value problem defined in terms of the Green identity and introduce the concept of Weyl operator function $M(\cdot)$ that agrees with other definitions found in the current literature. In typical cases of problems arising from the multidimensional partial equations of mathematical physics the function $M(\cdot)$ takes values in the set of unbounded densely defined operators acting on the auxiliary boundary space. Exact formulae are obtained and essential properties of $M(\cdot)$ are studied. In particular, we consider boundary problems defined by various boundary conditions and justify the well known procedure that reduces such problems to the 'equation on the boundary' involving the Weyl function, prove an analogue of the Borg-Levinson theorem, and link our results to the classical theory of extensions of symmetric operators.