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Approximation of magnetic Schrödinger operators with $δ$-interactions supported on networks

Markus Holzmann

TL;DR

The paper proves that a magnetic Schrödinger operator with a singular $\delta$-interaction supported on a finite network $\Sigma$ can be approximated in the norm resolvent sense by regular Schrödinger operators with scaled potentials supported in tubular neighborhoods of $\Sigma$. The δ-interaction is defined via a trace-based quadratic form with strength $\alpha\in L^p(\Sigma)+L^\infty(\Sigma)$, and the approximating potentials $V_\varepsilon$ are constructed from regular data $V^{(k)}$. A key technical step is a trace-shift estimate comparing function traces on $\Sigma$ and on nearby shifted surfaces, together with a KLMN-type argument, yielding a rate $\varepsilon^{\gamma_1/2}$ for the norm-resolvent convergence. As a consequence, spectra of the regularized operators converge to those of the singular model, enabling spectral-transfer results for complex potentials and network geometries, including graphs and piecewise $C^2$ boundaries, with implications for leaky quantum graphs and related applications.

Abstract

This paper deals with the approximation of a magnetic Schrödinger operator with a singular $δ$-potential that is formally given by $(i \nabla + A)^2 + Q + αδ_Σ$ by Schrödinger operators with regular potentials in the norm resolvent sense. This is done for $Σ$ being the finite union of $C^2$-hypersurfaces, for coefficients $A$, $Q$, and $α$ under almost minimal assumptions such that the associated quadratic forms are closed and sectorial, and $Q$ and $α$ are allowed to be complex-valued functions. In particular, $Σ$ can be a graph in $\mathbb{R}^2$ or the boundary of a piecewise $C^2$-domain. Moreover, spectral implications of the mentioned convergence result are discussed.

Approximation of magnetic Schrödinger operators with $δ$-interactions supported on networks

TL;DR

The paper proves that a magnetic Schrödinger operator with a singular -interaction supported on a finite network can be approximated in the norm resolvent sense by regular Schrödinger operators with scaled potentials supported in tubular neighborhoods of . The δ-interaction is defined via a trace-based quadratic form with strength , and the approximating potentials are constructed from regular data . A key technical step is a trace-shift estimate comparing function traces on and on nearby shifted surfaces, together with a KLMN-type argument, yielding a rate for the norm-resolvent convergence. As a consequence, spectra of the regularized operators converge to those of the singular model, enabling spectral-transfer results for complex potentials and network geometries, including graphs and piecewise boundaries, with implications for leaky quantum graphs and related applications.

Abstract

This paper deals with the approximation of a magnetic Schrödinger operator with a singular -potential that is formally given by by Schrödinger operators with regular potentials in the norm resolvent sense. This is done for being the finite union of -hypersurfaces, for coefficients , , and under almost minimal assumptions such that the associated quadratic forms are closed and sectorial, and and are allowed to be complex-valued functions. In particular, can be a graph in or the boundary of a piecewise -domain. Moreover, spectral implications of the mentioned convergence result are discussed.

Paper Structure

This paper contains 7 sections, 13 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The space $\mathcal{H}_{A, Q}$
  4. Description of hypersurfaces and their tubular neighborhoods
  5. Rigorous definition of $H_{A,Q,\alpha}$
  6. Proof of Theorem \ref{['theorem_approximation']}
  7. Consequences for the spectrum of $H_{A,Q,V_\varepsilon}$

Key Result

Theorem 1.1

Let $A$ and $Q = Q_1 +Q_2$ be as in assumption_A and assumption_Q, respectively, $\Sigma$ be as in Sigma_intro, and $V^{(k)} \in L^p(\mathbb{R}^n) + L^\infty(\mathbb{R}^n)$ with $p = \frac{n-1}{1-\gamma}$ for a $\gamma \in (0,1)$ be supported in $\Omega_{\beta}^{(k)}$. Let $V_\varepsilon$ be defined Eventually, let $H_{A, Q, \alpha}$ be the $m$-sectorial operator that is associated with the form i

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Example 2.6
  • ...and 18 more