Eigenvalues of the discrete Schrödinger operator in the large coupling constant limit
Siyu Gao
TL;DR
The paper analyzes eigenvalue crossings through a spectral gap for the discrete Schrödinger operator $A(\alpha)=A-\alpha V$ on $\mathbb Z^d$, with $A=-\Delta+f$ and a decaying impurity $V(n)\sim \Psi(\theta)|n|^{-p}$. Using the Birman-Schwinger principle and a spatial splitting into near, intermediate, and far regions, it reduces the problem to the spectrum of compact operators and a density-of-states integral. The main result is $N(\lambda,\alpha)\sim \alpha^{d/p}\int_{\mathbb R^d}(\rho(\lambda+\Psi(\theta)|x|^{-p})-\rho(\lambda))\,dx$ for $\lambda$ in the gap $\Lambda$, highlighting the roles of the decay exponent $p$ and dimension $d$. This extends continuous-space results to the lattice setting, linking impurity decay to spectral flow in periodic lattices and informing impurity-induced bound states and optical properties in crystals.
Abstract
Let $(λ_-,λ_+)$ be a spectral gap of a periodic Schrödinger operator $A$ on the lattice ${\mathbb Z}^d$. Consider the operator $A(α)=A-αV$ where $V$ is a decaying positive potential on ${\mathbb Z}^d$. We study the asymptotic behavior of the number of eigenvalues of $A(t)$ passing through a point $λ\in (λ_-,λ_+)$ as $t$ grows from $0$ to $α$.