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On the asymptotic behavior of the spectral gap for discrete Schrödinger operators

Matthias Hofmann, Joachim Kerner, Maximilian Pechmann

TL;DR

The work analyzes the asymptotic behavior of the spectral gap Γ(V_k, α) for discrete Schrödinger operators on a path graph as the volume grows, extending prior findings to a broad class of nonnegative, compactly supported potentials and solving a conjecture. By combining minmax principles with carefully constructed trial states and explicit eigenfunction analysis, the authors derive universal upper and lower bounds for the two smallest eigenvalues, linking them to Laplacian eigenvalues on subgraphs and to the potential’s support and strength. A key result is that, for any nonzero compactly supported potential, the gap decays faster than in the Laplacian case, with |V_k|^{2+η} Γ(V_k, α) → 0 for 0 ≤ η < 1, while in the special J = {0} case they obtain a precise limit lim_{k→∞} |V_k|^3 Γ(V_k, α) = 8π^2/α. This provides rigorous justification for observed numerical behavior and highlights a strong, nonlocal influence of localized potentials on infinite-volume spectral properties.

Abstract

In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schrödinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class of potentials using entirely different methods. Notably, we also resolve a conjecture previously proposed in this context. This then yields new insights into the rate at which the spectral gap tends to zero as the volume increases.

On the asymptotic behavior of the spectral gap for discrete Schrödinger operators

TL;DR

The work analyzes the asymptotic behavior of the spectral gap Γ(V_k, α) for discrete Schrödinger operators on a path graph as the volume grows, extending prior findings to a broad class of nonnegative, compactly supported potentials and solving a conjecture. By combining minmax principles with carefully constructed trial states and explicit eigenfunction analysis, the authors derive universal upper and lower bounds for the two smallest eigenvalues, linking them to Laplacian eigenvalues on subgraphs and to the potential’s support and strength. A key result is that, for any nonzero compactly supported potential, the gap decays faster than in the Laplacian case, with |V_k|^{2+η} Γ(V_k, α) → 0 for 0 ≤ η < 1, while in the special J = {0} case they obtain a precise limit lim_{k→∞} |V_k|^3 Γ(V_k, α) = 8π^2/α. This provides rigorous justification for observed numerical behavior and highlights a strong, nonlocal influence of localized potentials on infinite-volume spectral properties.

Abstract

In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schrödinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class of potentials using entirely different methods. Notably, we also resolve a conjecture previously proposed in this context. This then yields new insights into the rate at which the spectral gap tends to zero as the volume increases.

Paper Structure

This paper contains 6 sections, 14 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Spectral estimates
  4. Main results I: General Case
  5. Main results II: Convergence in the Special Case $J = \{0\}$
  6. Auxiliary results

Key Result

Theorem 3

Consider a Hamiltonian $H_{k,\boldsymbol{\alpha}}$ with ground state energy $\lambda_0(V_k, \boldsymbol \alpha)$. Then, for all $k > \max\{|r_{\min}|, |r_{\max}|\}/2$, we have Moreover, for all but finitely many $k \in \mathds N$.

Theorems & Definitions (30)

  • Remark 1
  • Remark 2
  • Theorem 3: Lower bound ground state energy
  • proof
  • Remark 4
  • Theorem 5: Upper bound ground state energy I
  • proof
  • Lemma 6: Bounds on first excited state energy
  • proof
  • Theorem 7: Asymptotic upper and lower bounds I
  • ...and 20 more