Bounds on the Pure Point Spectrum of Lattice Schrödinger Operators

Abstract

In dimension $d\ge 3$, a variational principle for the size of the pure point spectrum of (discrete) Schrödinger 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 $\lambda$.

arXiv comment24 pages

References in corpus (1)

• Note on the spectrum of discrete Schrödinger operators

Cited by in corpus (1)

• Uniform bounds of discrete Birman-Schwinger operators