Perturbation theory for spectral gap edges of 2D periodic Schrödinger operators

Abstract

We consider a two-dimensional periodic Schrödinger operator H = −∆+W with Γ being the lattice of periods. We investigate the structure of the edges of open gaps in the spectrum of H. We show that under arbitrary small perturbation V periodic with respect to N Γ where N = N (W ) is some integer, all edges of the gaps in the spectrum of H + V which are perturbation of the gaps of H become non-degenerate, i.e. are attained at finitely many points by one band function only and have non-degenerate quadratic minimum/maximum. We also discuss this problem in the discrete setting and show that changing the lattice of periods may indeed be unavoidable to achieve the non-degeneracy.

On spectral bands of discrete periodic operators

“…In the tight binding setting, the equivalent of [5] is false with the same counterexample as (2): the checkerboard potential. Some positive results (both in the discrete and continuous setting for d = 2) have been obtained in [16]: it is shown that one can construct a sequence of small perturbations of larger and larger periods to lower the degeneracy of the band edge and ultimately thansform the band function into a Morse function (however, one cannot treat all bands simultaneously by this method, as their number grows each time a perturbation is applied). One should note that (3) is a part of the "effective mass conjecture" which states that for "generic" potentials all band functions behave like Morse functions around their global minima and maxima.…”

“…In the case d = 2, Theorem 1.1 is sharp: the p-periodic potentials also produce one-dimensional Fermi surfaces at the edges of some bands. A simple argument has been provided in [16].…”

On the structure of band edges of $2$-dimensional periodic elliptic operators

“…An immediate consequence of our result is that Liouville theorems (in the sense of [17,18]) hold for the operator (2.1) at all gap edges, see Corollary 2.2. Our result can also be used in studying Green's function asymptotics near spectral gap edges, see [10,19,9], and to obtain a "variable period" version of the non-degeneracy conjecture in 2D [20].…”

Degenerate band edges in periodic quantum graphs

“…The well-known result in [22] established the validity of the full conjecture for the bottom of the spectrum of a periodic Schrödinger operator in Euclidean spaces, however the full conjecture still remains unproven for internal edges. It is worth mentioning that in the two dimensional situation, a "variable period" version of the non-degeneracy conjecture was found in [32] and the isolated nature of extrema for a wide class of Z 2 -periodic elliptic operators was recently established in [18]. In the discrete graph situation, the statement of the conjecture fails for periodic Schrödinger operators on a diatomic lattice (see [18]).…”