Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators

TL;DR

This paper investigates how eigenvalues and eigenvectors of unbounded operators with compact resolvents depend smoothly or analytically on parameters within various Denjoy-Carleman classes, extending classical perturbation theory.

Contribution

It establishes $C^M$-dependence results for eigenvalues and eigenvectors of unbounded operators with compact resolvents across a broad range of regularity classes, including analytic and quasianalytic classes.

Findings

Eigenvalues depend $C^M$-smoothly on parameters.

Eigenvectors depend $C^M$-smoothly on parameters.

Results encompass real analytic, quasianalytic, non-quasianalytic, and H"older classes.

Abstract

Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for $C^\om$ (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a H\"older continuity class $C^{0,\al}$. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.