Large Coupling Convergence Beyond Definiteness
Christian Koke
TL;DR
This work studies the asymptotic behavior of the operator family $A_\beta = A + \beta B$ as the coupling $\beta \to \infty$ without assuming positivity of $A$ or $B$. It develops a resolvent-based framework, avoiding form methods, to show that the limit is an effective operator defined on $\ker(B)$ via compression, with two principal regimes: strong resolvent convergence when $A$ and $B$ are self-adjoint and the compression of $A$ onto $\ker(B)$ is well-behaved, and norm resolvent convergence in the more general non-self-adjoint setting under spectral conditions (0 isolated in $\sigma(B)$ and vanishing quasi-nilpotent part) aided by the Riesz projector. The paper further dissects several subcases (relative boundedness of $A$ by $B$, bounded $B$, and block-operator/Schur-complement analyses) to establish explicit convergence rates and limit operators, highlighting that the limit can depend on the exact form of the Riesz projector when $B$ is not self-adjoint. By providing a suite of examples from PDEs, quantum physics, and graph theory, the results extend large-coupling perturbation theory beyond positivity and clarify how spectral structure governs effective reduced models. The findings have broad implications for deriving reduced descriptions in complex systems where traditional form-based methods fail or are inapplicable.
Abstract
We study convergence of operator families of the form $A_β= A + βB$ towards an effective operator defined on $\ker(B)$, as the coupling constant $β$ 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 $\{A_β\}_β$. Our findings are that: (i) \emph{Strong} resolvent convergence holds (without further spectral assumptions) if $A + βB$ is self-adjoint and the compression of $A$ onto $\ker(B)$ is well behaved. (ii) Under the more detailed assumption that $0 \in σ(B)$ is isolated, \emph{norm} resolvent convergence can be established even if $A+βB$ 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 $0 \in σ(B)$ onto $\ker(B)$.