Exact Results in Quantum Theory

The Schrödinger equation describes the time evolution of quantum states as elements of a suitable Hilbert space. For this reason, the continuous equation on L^2(R^d) has been extensively studied, especially with regard to the behaviour of the eigenvalues and eigenfunctions of the associated Schrödinger operator as the physical parameters vary. By contrast, much less is known about the discrete analogue, despite its fundamental role in numerical analysis and in lattice models from mathematical physics. In this talk, I will discuss the asymptotic behaviour of the eigenvalues of the operator H_N = −N^2⁄2\Delta + λ_NV_N acting on l^2(ℤ^d). Here, the parameter N governs the continuum limit, while λN is related to a semiclassical regime in which \hbar \rightarrow 0. In particular, I will identify a regime for the semiclassical parameter in which the eigenvalues of the discrete Schrödinger operator converge to those of an associated continuous harmonic oscillator in the N \rightarrow+\infty limit. I will also show that, in certain parameter regimes, the behaviour of H_N differs significantly from its continuous counterpart. As a byproduct, I will also present a new technique for deriving upper and lower bounds for all energy levels of a class of Schrödinger operators that arises naturally in the analysis. The talk is based on joint work (arXiv:2602.23156) with Christian van de Ven and Matthias Keller.