Bounds on the Discrete Spectrum of Lattice Schrödinger Operators

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) Schrödinger operators on the $d$--dimensional, $d\ge 1$, cubic lattice ${\mathbb{Z}}^d$ at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension $d\ge 1$ -- even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions $d\ge 1$ 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 $d\ge 3$, 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 Schrödinger operators if $d\ge 3$. We show here how to construct general upper bounds on the number of bound states of Schrödinger operators on ${\mathbb{Z}}^d$ from semi-classical quantities in all space dimensions $d\ge 1$ and independently of the positivity-improving property of the free Hamiltonian.

33 pages

Cited by in corpus (2)

• Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions
• On the number and sums of eigenvalues of Schrödinger-type operators with degenerate kinetic energy