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A Schrödinger operator with confining potential having quadratic growth

Chiara Alessi, Lorenzo Brasco, Michele Miranda

TL;DR

This work analyzes a Schrödinger operator with a confining quadratic growth potential V(x)=(dist(x,Σ))^2, where Σ is a compact set. By formulating the problem variationally and normalizing to the unit-scale case, the authors establish a discrete spectrum, provide explicit ground-state bounds, and derive stability estimates for λ_n under Hausdorff perturbations of the well. They prove strong regularity and summability properties of eigenstates, including global L^∞ bounds, weighted integrability, and higher regularity up to C^{1,α}, along with explicit exponential decay at infinity. Finally, they quantify how eigenspaces converge to those of the quantum harmonic oscillator as Σ shrinks to the origin, yielding precise L^2 and H^1-stability results that illuminate the impact of geometric confinement on spectral data.

Abstract

We study the spectral properties of a Schrödinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates, lower bounds on the ground state energy, regularity and integrability properties of eigenstates. We also get explicit decay estimates at infinity, by means of elementary nonlinear methods.

A Schrödinger operator with confining potential having quadratic growth

TL;DR

This work analyzes a Schrödinger operator with a confining quadratic growth potential V(x)=(dist(x,Σ))^2, where Σ is a compact set. By formulating the problem variationally and normalizing to the unit-scale case, the authors establish a discrete spectrum, provide explicit ground-state bounds, and derive stability estimates for λ_n under Hausdorff perturbations of the well. They prove strong regularity and summability properties of eigenstates, including global L^∞ bounds, weighted integrability, and higher regularity up to C^{1,α}, along with explicit exponential decay at infinity. Finally, they quantify how eigenspaces converge to those of the quantum harmonic oscillator as Σ shrinks to the origin, yielding precise L^2 and H^1-stability results that illuminate the impact of geometric confinement on spectral data.

Abstract

We study the spectral properties of a Schrödinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates, lower bounds on the ground state energy, regularity and integrability properties of eigenstates. We also get explicit decay estimates at infinity, by means of elementary nonlinear methods.
Paper Structure (12 sections, 20 theorems, 280 equations)

This paper contains 12 sections, 20 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Preliminaries
  4. Notation and basics
  5. Properties of the potentials
  6. Spectral properties
  7. Stability of the spectrum
  8. Summability of eigenstates
  9. Higher regularity of eigenstates
  10. Exponential decay at infinity
  11. Stability of eigenspaces
  12. The case of $\mathbb{S}^1$ in $\mathbb{R}^3$

Key Result

Lemma 2.1

The potential $V$ is a locally Lipschitz function such that we have and where $\delta,R$ are defined in deltaR. In particular, we have

Theorems & Definitions (50)

  • Example 1.1
  • Remark 1.2: Reduction to the case $\omega=1$
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 40 more