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Discrete spectrum from local perturbations of leaky curves

Pavel Exner

TL;DR

This work analyzes a singular Schrödinger operator with δ-type interaction supported on a curve (a leaky-curve) and investigates how local perturbations of a periodic interaction support can generate discrete bound states below the essential spectrum. By combining Floquet theory for the periodic baseline with a singular Birman-Schwinger principle, the authors derive criteria under which local contractions of the interaction region produce eigenvalues below the bottom of the essential spectrum, including a delicate zero-mean (critical) case. They establish essential-spectrum stability under local perturbations and demonstrate, via a constructed Birman-Schwinger analysis and mollified trial states, that discrete spectrum can appear even for non-small perturbations, with strong-coupling asymptotics linking to a one-dimensional effective curvature potential. The results deepen understanding of geometry-induced spectral effects in lower-dimensional quantum systems and provide concrete methods to detect and estimate bound states arising from local geometric perturbations of leaky quantum wires.

Abstract

We discuss spectrum of a class of singular Schrödinger operator models known as leaky curves and show that if the interaction support has a periodic shape, its local perturbations can give rise to a discrete spectrum below the continuum threshold even if they are of `zero mean'.

Discrete spectrum from local perturbations of leaky curves

TL;DR

This work analyzes a singular Schrödinger operator with δ-type interaction supported on a curve (a leaky-curve) and investigates how local perturbations of a periodic interaction support can generate discrete bound states below the essential spectrum. By combining Floquet theory for the periodic baseline with a singular Birman-Schwinger principle, the authors derive criteria under which local contractions of the interaction region produce eigenvalues below the bottom of the essential spectrum, including a delicate zero-mean (critical) case. They establish essential-spectrum stability under local perturbations and demonstrate, via a constructed Birman-Schwinger analysis and mollified trial states, that discrete spectrum can appear even for non-small perturbations, with strong-coupling asymptotics linking to a one-dimensional effective curvature potential. The results deepen understanding of geometry-induced spectral effects in lower-dimensional quantum systems and provide concrete methods to detect and estimate bound states arising from local geometric perturbations of leaky quantum wires.

Abstract

We discuss spectrum of a class of singular Schrödinger operator models known as leaky curves and show that if the interaction support has a periodic shape, its local perturbations can give rise to a discrete spectrum below the continuum threshold even if they are of `zero mean'.
Paper Structure (7 sections, 9 theorems, 59 equations, 1 figure)

This paper contains 7 sections, 9 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The singular version of Birman-Schwinger principle
  4. The perturbed essential spectrum
  5. Discrete spectrum due to local perturbations
  6. The perturbation may not be gentle
  7. Concluding remarks

Key Result

Proposition 2.1

Under the assumption ai, the spectrum of operator $H_{\alpha,\Gamma_0}$ is purely essential, with $\inf \sigma(H_{\alpha,\Gamma_0})<0$; the negative part is absolutely continuous.

Figures (1)

  • Figure 1: The interaction support $\Gamma_\tau$ in the example with one curved segment shifted

Theorems & Definitions (20)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 5.1
  • ...and 10 more