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On two properties of positively perturbed discrete Schrödinger operators

Ognjen Milatovic

TL;DR

This work analyzes positive perturbations of lower semi-bounded discrete Schrödinger operators on weighted graphs under the finiteness condition FC. It proves two main results: first, the form-sum of a Friedrichs extension with a non-negative multiplication operator coincides with the Friedrichs extension of the perturbed operator, $L_{V_1}\widetilde{+}M_{V_2}=L_V$; second, essential self-adjointness is stable under non-negative perturbations when the perturbation is controlled relative to the unperturbed operator, all without assuming metric completeness. The approach relies on Green's formula, Beurling–Deny properties, the positive form core for Friedrichs extensions, and resolvent convergence via truncation alongside Kato-type arguments. These results extend continuum intuitions to discrete graphs, providing robust tools for form and operator extensions of Schrödinger-type operators in non-locally finite graph settings. The findings have potential implications for spectral theory, self-adjointness criteria, and the analysis of quantum dynamics on graphs.

Abstract

We show that if we start from a symmetric lower semi-bounded Schrödinger operator $\mathcal{H}$ on finitely supported functions on a discrete weighted graph (satisfying certain conditions), apply the Friedrichs construction to get a self-adjoint extension $H$, and then perturb $H$ by a non-negative function $W$, then the resulting form-sum $H\widetilde{+}W$ coincides with the Friedrichs extension of $\mathcal{H}+W$. Additionally, we consider a non-negative perturbation of an essentially self-adjoint lower semi-bounded Schrödinger operator $H$ on a discrete weighted graph. We show that, under certain conditions on the graph and the perturbation, the essential self-adjointness of $H$ remains stable under the given perturbation.

On two properties of positively perturbed discrete Schrödinger operators

TL;DR

This work analyzes positive perturbations of lower semi-bounded discrete Schrödinger operators on weighted graphs under the finiteness condition FC. It proves two main results: first, the form-sum of a Friedrichs extension with a non-negative multiplication operator coincides with the Friedrichs extension of the perturbed operator, ; second, essential self-adjointness is stable under non-negative perturbations when the perturbation is controlled relative to the unperturbed operator, all without assuming metric completeness. The approach relies on Green's formula, Beurling–Deny properties, the positive form core for Friedrichs extensions, and resolvent convergence via truncation alongside Kato-type arguments. These results extend continuum intuitions to discrete graphs, providing robust tools for form and operator extensions of Schrödinger-type operators in non-locally finite graph settings. The findings have potential implications for spectral theory, self-adjointness criteria, and the analysis of quantum dynamics on graphs.

Abstract

We show that if we start from a symmetric lower semi-bounded Schrödinger operator on finitely supported functions on a discrete weighted graph (satisfying certain conditions), apply the Friedrichs construction to get a self-adjoint extension , and then perturb by a non-negative function , then the resulting form-sum coincides with the Friedrichs extension of . Additionally, we consider a non-negative perturbation of an essentially self-adjoint lower semi-bounded Schrödinger operator on a discrete weighted graph. We show that, under certain conditions on the graph and the perturbation, the essential self-adjointness of remains stable under the given perturbation.

Paper Structure

This paper contains 13 sections, 15 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Weighted Graphs and Basic Function Spaces
  4. Finiteness Condition (FC)
  5. Discrete Schrödinger Operators
  6. Statements of the results
  7. Preliminaries
  8. Green's Formula
  9. Quadratic Forms and Self-adjoint Operators
  10. Positive Form Core
  11. Preliminary Lemmas
  12. Proof of Theorem \ref{['T:main-1']}
  13. Proof of theorem \ref{['T:main-2']}

Key Result

Theorem 2.1

Let $(X, b, \mu)$ be a weighted and connected graph satisfying the property (FC) as in section SS:FC. Assume that $V=V_1+V_2$, where $V_j\in C(X)$ are real-valued, $V_2(x)\geq 0$ for all $x\in X$, and $V_1$ satisfies the condition (E:l-v-1). Let $L_{V_1}$ and $L_{V}$ be the Friedrichs extensions of

Theorems & Definitions (28)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Remark 3.5
  • Corollary 3.6
  • proof
  • ...and 18 more