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