On spectral band edges of discrete periodic Schrodinger operators

Abstract: We consider discrete Schrodinger operators on $\ell^2(\mathbb Z^d)$, periodic with respect to some lattice $\Gamma$ in $\mathbb Z^d$ of full rank. Our main goal is to study dimensions of level sets of spectral band functions at the energies corresponding to their extremal values (the edges of the bands).Suppose that $d\ge 3$ and the dual lattice $\Gamma’$ does not contain the vector $(1/2,…,1/2)$. Then the above mentioned level sets have dimension at most $d-2$.

Suppose that $d=2$ and the dual lattice does not contain vectors of the form $(1/p,1/p)$ and $(1/p,-1/p)$ for all $p\ge 2$. Then the same statement holds (in other words, the corresponding level sets are finite modulo $\mathbb Z^d$).For all lattices that do not satisfy the above assumptions, there are known counterexamples of level sets of dimensions $d-1$.

Part of the argument also implies a discrete Bethe-Sommerfeld property: if $d\ge 2$ and the dual lattice does not contain the vector $(1/2,…,1/2)$, then, for sufficiently small potentials (depending on the lattice), the spectrum of the periodic Schrodinger operator is an interval. Previously, this property was studied by Kruger, Embree-Fillman, Jitomirskaya-Han, and Fillman-Han. Our proof is different and implies some new cases.

The talk is based on joint work with in progress with N. Filonov.

mathematical physics

Audience: researchers in the topic

TAMU: Mathematical Physics and Harmonic Analysis Seminar