Boundary triplets and the index of families of self-adjoint elliptic boundary problems

TL;DR

This paper develops an abstract framework for boundary triplets to analyze the index of families of self-adjoint elliptic boundary problems, providing new proofs and index difference formulas.

Contribution

It introduces an axiomatic approach to boundary triplets and applies it to derive index theorems for self-adjoint elliptic boundary problems.

Findings

Provides an analytic proof of the index theorem for Dirac-like boundary problems

Derives an Agranovich-Dynin type formula for index differences

Establishes an axiomatic boundary triplet construction

Abstract

The paper is devoted to an abstract axiomatic version of a construction of boundary triplets implicit in the works of M.I. Vishik and G. Grubb and its applications to the index of families of self-adjoint elliptic differential boundary problems of order one. This leads to an analytic proof of the index theorem for Dirac-like self-adjoint boundary problems from arXiv:2207.09574, and to an Agranovich-Dynin type theorem computing the difference of indices of families of self-adjoint boundary problems differing only by the boundary conditions.