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Coupling of the continuum and semiclassical limit. Part I: convergence of eigenvalues

Matthias Keller, Lorenzo Pettinari, Christiaan J. F. van de Ven

Abstract

We analyze the semiclassical $d$-dimensional Schrödinger operator in the continuum $ \frac{1}{2} Δ+ λ_N^2 V$ discretized on a mesh with spacing proportional to $1/N$. The semi-classical parameter $λ_N$ is chosen as $λ_N = N^{1 - γ}$, with $γ\in (-1,1)$, which ensures that $N$ governs both the semiclassical and continuum limit simultaneously. We prove that all eigenvalues of the discrete operator converge to those of the continuum, as $λ_N\to\infty$. Beyond this semi-classical domain, in the case of the harmonic oscillator, we further discuss the spectral asymptotics for $γ\in \mathbb{R} \setminus (-1,1)$, thereby fully characterizing the eigenvalue behavior across all possible values of $γ\in\mathbb{R}$.

Coupling of the continuum and semiclassical limit. Part I: convergence of eigenvalues

Abstract

We analyze the semiclassical -dimensional Schrödinger operator in the continuum discretized on a mesh with spacing proportional to . The semi-classical parameter is chosen as , with , which ensures that governs both the semiclassical and continuum limit simultaneously. We prove that all eigenvalues of the discrete operator converge to those of the continuum, as . Beyond this semi-classical domain, in the case of the harmonic oscillator, we further discuss the spectral asymptotics for , thereby fully characterizing the eigenvalue behavior across all possible values of .
Paper Structure (12 sections, 20 theorems, 141 equations, 1 figure)

This paper contains 12 sections, 20 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Semiclassical analysis and discrete Schrödinger operators
  3. Physical interpretation and comparison with literature
  4. Main results on eigenvalue asymptotics
  5. Discrete harmonic oscillators
  6. Eigenvalue upper bound for harmonic oscillators
  7. Eigenvalue upper bound for modified harmonic oscillators
  8. IMS localization formula
  9. General potentials
  10. Eigenvalue upper bound for general potentials
  11. Eigenvalue lower bound for general potentials
  12. Alternative scaling regimes

Key Result

Theorem 1.2

Let $V$ be a potential satisfying Assumption assumption and $\Sigma(V)=\{e_0(V)\leq e_1(V)\leq\ldots\leq e_n(V)\leq \ldots\}$ as above. Then, for all $n\in \mathbb{N}_0$ and $\gamma\in (-1,1)$, it holds where $\lambda_N=N^{1-\gamma}$.

Figures (1)

  • Figure 1: Dependence on $\gamma$

Theorems & Definitions (42)

  • Remark
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5: Upper bound
  • proof
  • proof : Proof of Proposition \ref{['thm:upperbound']}
  • ...and 32 more