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

TL;DR

This paper proves the convergence of eigenvalues of a discretized Schrödinger operator to the continuum limit in a semi-classical setting, and explores spectral asymptotics beyond this regime for the harmonic oscillator.

Contribution

It establishes the eigenvalue convergence for a discretized Schrödinger operator under a specific semi-classical scaling and characterizes spectral behavior for all scaling regimes.

Findings

Eigenvalues of the discretized operator converge to the continuum eigenvalues as the semi-classical parameter grows.

Spectral asymptotics are fully characterized for the harmonic oscillator across all semi-classical regimes.

The analysis bridges continuum and semi-classical limits in quantum spectral theory.

Abstract

We analyze the semiclassical $d$-dimensional Schr\"{o}dinger operator in the continuum $ \frac{1}{2} \Delta + \lambda_N^2 V$ discretized on a mesh with spacing proportional to $1/N$. The semi-classical parameter $\lambda_N$ is chosen as $\lambda_N = N^{1 - \gamma}$, with $\gamma \in (-1,1)$, 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 $\lambda_N\to\infty$. Beyond this semi-classical domain, in the case of the harmonic oscillator, we further discuss the spectral asymptotics for $\gamma \in \mathbb{R} \setminus (-1,1)$, thereby fully characterizing the eigenvalue behavior across all possible values of $\gamma\in\mathbb{R}$.