First-order asymptotic perturbation theory for extensions of symmetric operators

TL;DR

This paper develops a new asymptotic perturbation theory for self-adjoint extensions of symmetric operators using symplectic methods, deriving formulas for resolvent expansions, eigenvalue bifurcations, and spectral flow, with applications in quantum graphs and PDEs.

Contribution

It introduces a symplectic framework for asymptotic analysis of operator extensions, including a new Krein formula and variational formulas for eigenvalues.

Findings

Derived a Riccati-type differential equation for resolvent asymptotics.

Established a symplectic version of the Kato selection theorem.

Applied theory to quantum graphs, PDEs, and heat equations.

Abstract

This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness we obtain a new version of Krein formula for resolvent difference which facilitates asymptotic analysis of resolvent operators via first order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati-type differential equation and the first order asymptotic expansion for resolvents of self-adjoint extensions determined by smooth one-parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard-Rellich-type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self-adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig-Penney model, elliptic second order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.