Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators

TL;DR

This paper extends the classical Q-function concept to elliptic differential operators, interpreting the Dirichlet-to-Neumann map as a generalized Q-function, and derives explicit formulas for resolvent differences and trace formulas.

Contribution

It introduces a generalized Q-function framework for elliptic operators, linking it to Dirichlet-to-Neumann maps and providing explicit Krein type formulas.

Findings

Derived explicit Krein type formulas for resolvent differences.

Established trace formulas within an H^2-framework.

Unified the theory of Q-functions with elliptic boundary value problems.

Abstract

The classical concept of $Q$-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized $Q$-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an $H^2$-framework are obtained.