Resolvent expansions for self-adjoint operators via boundary triplets

TL;DR

This paper uses boundary triplet theory to analyze how small domain variations affect self-adjoint operators, providing second-order asymptotics for resolvents, spectral projections, and eigenvalues, with explicit formulas and applications.

Contribution

It introduces a second-order perturbation framework for self-adjoint operators using boundary triplets, including explicit formulas for eigenvalue variations and applications to Robin Laplacians.

Findings

Explicit formulas for first variation and Hessian of eigenvalues.

Second-order asymptotic analysis of resolvents and spectral projections.

Application to Robin Laplacian and extensions.

Abstract

In this paper we develop certain aspects of perturbation theory for self-adjoint operators subject to small variations of their domains. We use the abstract theory of boundary triplets to quantify such perturbations and give the second order asymptotic analysis for resolvents, spectral projections, discrete eigenvalues of the corresponding self-adjoint operators. In particular, we derive explicit formulas for the first variation and the Hessian of the eigenvalue curves bifurcating from a discrete eigenvalue of an unperturbed operator. An application is given to a matrix valued Robin Laplacian and more general Robin-type self-adjoint extensions.