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A survey on the resolvent convergence

Joaquim Duran

TL;DR

This survey analyzes how the resolvent of unbounded self-adjoint operators encodes spectral information and provides robust notions of convergence (norm, strong, and weak resolvent convergence) that remain well-defined for operators with varying domains. It systematically relates resolvent convergence to the convergence of spectra and situates these notions within the broader frameworks of strong/weak graph limits and $G$- and $ ext{Γ}$-convergence of quadratic forms, including precise equivalences under compact embedding. A central thread is that norm resolvent convergence yields full spectral convergence (including essential/discrete parts), while strong resolvent convergence may cause spectral contraction, with the Robin Laplacian serving as a concrete exemplar where degenerating boundary conditions lead to Neumann or Dirichlet limits in the norm resolvent sense. The chapter also furnishes a diagram of implications and demonstrates the theory with a detailed elliptic-operator example, highlighting practical pathways to analyze spectral stability under operator convergence. Overall, the work clarifies how resolvent-based frameworks unify spectral convergence with variational and graph-limit approaches in complex Hilbert spaces.

Abstract

This chapter deals with the notion of the resolvent of a self-adjoint operator. We pay special attention to the convergence of unbounded self-adjoint operators in several resolvent senses, and how they are related to the convergence of their spectra. We also explore the relations that these notions of convergence have with the so-called strong graph limit, $G$-convergence, and $Γ$-convergence.

A survey on the resolvent convergence

TL;DR

This survey analyzes how the resolvent of unbounded self-adjoint operators encodes spectral information and provides robust notions of convergence (norm, strong, and weak resolvent convergence) that remain well-defined for operators with varying domains. It systematically relates resolvent convergence to the convergence of spectra and situates these notions within the broader frameworks of strong/weak graph limits and - and -convergence of quadratic forms, including precise equivalences under compact embedding. A central thread is that norm resolvent convergence yields full spectral convergence (including essential/discrete parts), while strong resolvent convergence may cause spectral contraction, with the Robin Laplacian serving as a concrete exemplar where degenerating boundary conditions lead to Neumann or Dirichlet limits in the norm resolvent sense. The chapter also furnishes a diagram of implications and demonstrates the theory with a detailed elliptic-operator example, highlighting practical pathways to analyze spectral stability under operator convergence. Overall, the work clarifies how resolvent-based frameworks unify spectral convergence with variational and graph-limit approaches in complex Hilbert spaces.

Abstract

This chapter deals with the notion of the resolvent of a self-adjoint operator. We pay special attention to the convergence of unbounded self-adjoint operators in several resolvent senses, and how they are related to the convergence of their spectra. We also explore the relations that these notions of convergence have with the so-called strong graph limit, -convergence, and -convergence.

Paper Structure

This paper contains 8 sections, 16 theorems, 57 equations, 1 figure.

Table of Contents

  1. Motivation for a complex setting
  2. Resolvent and spectrum of an operator
  3. Resolvent convergence
  4. Convergence of spectra
  5. Relation with strong graph limit
  6. Relation with $G$- and $\Gamma$-convergence
  7. Diagram summarizing the resolvent convergence
  8. Example: resolvent convergence of the Robin Laplacian

Key Result

theorem 1

Let $A$ be a self-adjoint operator. The resolvent set $\rho(A)$ is a non-empty open subset of $\mathbb C$ on which the map $\rho(A)\ni \lambda\mapsto R_\lambda(A)$ is analytic. That is, $R_\lambda(A)$ has an absolutely convergent power series expansion around every $\lambda\in \rho(A)$. Moreover, th

Figures (1)

  • Figure 1: Diagram of the relations between the different notions of convergence of densely defined, self-adjoint operators $A_n$, $n\in\mathbb N \cup\{\infty\}$, in a separable, complex Hilbert space $(H,\langle\cdot,\cdot\rangle, \|\cdot\|)$.

Theorems & Definitions (31)

  • definition 1
  • definition 2
  • definition 3
  • remark 1
  • definition 4
  • theorem 1
  • definition 5
  • theorem 2
  • corollary 1
  • definition 6
  • ...and 21 more