Krein's Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators

TL;DR

This paper derives a Krein-type formula to explicitly relate the resolvents of different self-adjoint extensions of a symmetric second order elliptic differential operator on a bounded domain, aiding spectral analysis.

Contribution

It introduces a Krein's resolvent formula specifically for the Friedrichs and other self-adjoint extensions of symmetric elliptic operators on bounded domains with smooth boundaries.

Findings

Explicit formula for resolvent difference between extensions

Facilitates spectral analysis of elliptic operators

Applicable to operators with $C^{1,1}$ boundary

Abstract

Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain with $C^{1,1}$ boundary, we provide a Krein-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.