Year: 2024
Type: article
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 use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that 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.
DOI: 10.1112/jlms.13005
Authors: Yuri Latushkin, Selim Sukhtaiev
Institutions: University of Missouri, Auburn University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G154770667, LEAPS-MPS:Research in Applied Analysis: Evolution Equations on Networks, Research in Applied Dynamical Systems, Conference: Joint Alabama--Florida Conference on Differential Equations, Dynamical Systems and Applications, Collaborative Research: Stability and Instability of Periodically Stationary Nonlinear Waves with Applications to Fiber Lasers
OpenAlex ID: W4403333250