Bounds on the Pure Point Spectrum of Lattice Schr\"odinger Operators

TL;DR

This paper establishes a variational principle for the pure point spectrum size of lattice Schrödinger operators in dimensions three and higher, providing conditions for absence of embedded eigenvalues and employing advanced spectral analysis techniques.

Contribution

It introduces a new variational principle for the pure point spectrum of lattice Schrödinger operators with Morse dispersion relations and decaying potentials, extending spectral theory insights.

Findings

Absence of embedded and threshold eigenvalues under specified conditions

A variational estimate for the pure point spectrum size

Application of Mourre estimate and Virial theorem in proof

Abstract

In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schr\"odinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $|x|^{-2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of $H(\mathfrak{e},V)$ for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator $B$ and positive number $\lambda >0$, the number of negative eigenvalues of $\lambda B$ is independent of $\lambda$.