Schrödinger operators on periodic discrete graphs

Abstract

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrödinger operators, constructed in the paper.

Magnetic Schrödinger operators on periodic discrete graphs

“…We note that for non-magnetic operators similar estimates were obtained for the Lebesgue measure of the spectrum in [KS14] and for effective masses in [KS16].…”

“…Items i) -v) for the case α = 0 were proved in [KS14] (Proposition 7.2). Combining (5.8) and (5.9) we obtain |σ(H α )| = 4β, i.e., the estimates (2.15) become identities.…”

“…In Section 5 we describe some simple properties of fiber magnetic Laplacians and Schrödinger operators and show that the spectral estimates obtained in Theorem 2.3 become identities for a specific graph. In the proof we use an example from [KS14]. In Section 5 we also recall some well-known properties of matrices needed to prove our main results.…”

Spectral band localization for Schrödinger operators on discrete periodic graphs

“…see [KS13], where κ j is the degree of v j , δ jk is the Kronecker delta and · , · denotes the standard inner product in R d . Now we need the following simple fact (see Theorem 4.3.1 in [HJ85]).…”

“…(2) Korotyaev and Saburova [KS13] considered Schrödinger operators on the discrete graphs and estimated the Lebesgue measure of their spectrum in terms of geometric parameters of the graph only.…”