Large Coupling Convergence Beyond Definiteness

TL;DR

This paper investigates the convergence of operator families of the form A+βB as β approaches infinity, extending classical results to cases where neither operator is semi-definite, using resolvent identities instead of form methods.

Contribution

It establishes strong and norm resolvent convergence results for operator families beyond the classical positive semi-definite setting, considering non-self-adjoint operators and spectral conditions.

Findings

Strong resolvent convergence without spectral assumptions

Norm resolvent convergence with spectral isolation of zero

Limit operator depends on Riesz projector when B is non-self-adjoint

Abstract

We study convergence of operator families of the form $A_\beta = A + \beta B$ towards an effective operator defined on $\ker(B)$, as the coupling constant $\beta$ tends to infinity. Crucially, we focus on the setting where neither $A$ nor $B$ can be assumed to be positive- (or negative-) semi-definite. We are hence outside the classical form-theoretic framework, where results based on Kato's monotone convergence theorem would be applicable. Thus, instead of form methods, our approach builds on classical resolvent identities to study convergence of the family $\{A_\beta\}_{\beta}$. Our findings are that: (i) \emph{Strong} resolvent convergence holds (without further spectral assumptions) if $A + \beta B$ is self-adjoint and the compression of $A$ onto $\ker(B)$ is well behaved. (ii) Under the more detailed assumption that $0 \in \sigma(B)$ is isolated, \emph{norm} resolvent convergence can be established even if $A+\beta B$ is merely closed, provided the quasinilpotent part of $B$ at zero vanishes and certain conditions on the interplay of $A$ and $B$ are met. Importantly, if $B$ is not self-adjoint we find that the limit operator not only depends on $\ker(B)$ as a Hilbert space, but crucially also on the precise form of the Riesz projector at $0 \in \sigma(B)$ onto $\ker(B)$.