Bounds on the Discrete Spectrum of Lattice Schr\"odinger Operators

TL;DR

This paper investigates the validity of Weyl asymptotics for the discrete spectrum of lattice Schrödinger operators, showing it can be violated or hold depending on decay conditions of the potential across different dimensions.

Contribution

It demonstrates conditions under which Weyl asymptotics are valid or violated for lattice Schrödinger operators, providing new bounds on the number of bound states in all dimensions.

Findings

Weyl asymptotics can be violated in any dimension at large couplings.

Weyl asymptotics always hold for potentials with mild decay conditions in dimensions d≥3.

Constructs general upper bounds on bound states from semi-classical quantities in all dimensions.

Abstract

We discuss the validity of the Weyl asymptotics -- in the sense of two-sided bounds -- for the size of the discrete spectrum of (discrete) Schr\"odinger operators on the $d$--dimensional, $d\geq 1$, cubic lattice $\mathbb{Z}^{d}$ at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension $d\geq 1$ -- even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions $d\geq 1$ that, for potentials well-behaved at infinity and fulfilling suitable decay conditions, the Weyl asymptotics always hold. These decay conditions are mild in the case $d\geq 3$, while stronger for $d=1,2$. It is well-known that the semi-classical number of bound states is -- up to a constant -- always an upper bound on the size of the discrete spectrum of Schr\"odinger operators if $d\geq 3$. We show here how to construct general upper bounds on the number of bound states of Schr\"odinger operators on $\mathbb{Z}^{d}$ from semi-classical quantities in all space dimensions $d\geq 1$ and independently of the positivity-improving property of the free Hamiltonian.