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

We analyze the semiclassical $d$-dimensional Schrödinger operator in the continuum $\frac{1}{2}\mathrm{\Delta }+{\lambda }_N^{2}V$ discretized on a mesh with spacing proportional to $1\mathrm{/}N$. 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 ${\lambda }_N\to \mathrm{\infty }$. 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 $\gamma \in \mathbb{R}$.