Dirichlet-to-Neumann Maps, Abstract Weyl-Titchmarsh $M$-Functions, and a Generalized Index of Unbounded Meromorphic Operator-Valued Functions

TL;DR

This paper develops a generalized index theory for certain unbounded operator-valued functions, linking spectral properties of non-self-adjoint Schrödinger operators and Dirichlet-to-Neumann maps through index formulas.

Contribution

It introduces a new generalized index for meromorphic operator-valued functions and establishes formulas connecting eigenvalue multiplicities with these indices.

Findings

Derived index formulas relating eigenvalues and Dirichlet-to-Neumann maps.

Extended index theory to non-self-adjoint Schrödinger operators.

Unified framework for Weyl-Titchmarsh functions and boundary maps.

Abstract

We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schr\"odinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh $M$-functions and Donoghue-type $M$-functions corresponding to closed extensions of symmetric operators belong to it. The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schr\"{o}dinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl-Titchmarsh $M$-functions, respectively.