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

Abstract

For a wide class of 2D periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.

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

“…6. The same conclusions will hold if we consider a more general class of unperturbed operators, say the periodic magnetic Schrödinger operators (or even periodic second order coefficients); effectively, the only property we need from a class of operators we consider is the finiteness of the set S, see [2].…”

“…Counter-example. The following examle is due to N. Filonov (see [2]). We define the discrete Schrödinger operator in l 2 (Z 2 ) as H = ∆ + V , where (∆u) (n 1 ,n 2 ) = u (n 1 +1,n 2 ) + u (n 1 −1,n 2 ) + u (n 1 ,n 2 +1) + u (n 1 ,n 2 −1) , and (V u) (n 1 ,n 2 ) = V 0 u (n 1 ,n 2 ) for n 1 + n 2 being even and (V u) (n 1 ,n 2 ) = V 1 u (n 1 ,n 2 ) for n 1 + n 2 being odd.…”

Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds

“…• The main results in this paper can be easily carried over to the case when the band edge occurs at finitely many quasimomenta k 0 in the Brillouin zone (instead of assuming the condition A3) by summing the asymptotics coming from all these non-degenerate isolated extrema. It was shown in [13] that for a wide class of two dimensional periodic secondorder elliptic operators (including the class of operators we consider in this paper and periodic magnetic Schrödinger operators in 2D), the extrema of any spectral band function (not necessarily spectral edges) are attained on a finite set of values of the quasimomentum in the Brillouin zone. • The proofs of the main results go through verbatim for periodic elliptic secondorder operators acting on vector bundles over the abelian covering X.…”

Gap localization of TE‐modes by arbitrarily weak defects

“…However, to show absence of bound states for periodic divergence type operators seems to be extremely difficult. For periodic operators with sufficiently smooth coefficients, this question is investigated and addressed in .…”