Spectral estimates for Schrödinger operators on periodic discrete graphs

Abstract

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.

Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs

“…and τ : A * → Z d is the index form defined by 1.8, 1.10. This number b depends essentially on the choice of the embedding of the periodic graph G into the space R d , i.e., b is not an invariant for G. Similar estimates for the normalized Laplacian ∆ n (see 2.11) were obtained in [18]:…”

“…Thus, the total bandwidth may exceed this number. Upper estimates of the total bandwidth for the Schrödinger operator with a periodic potential in terms of geometric parameters of the graph were obtained in [14,18,19]. For Schrödinger operators with periodic magnetic potentials similar upper estimates were obtained in [17].…”

“…We remark that the last spectral band σ ν (H) of the Schrödinger operator H = −∆ + V on periodic graphs is non-degenerate, see [18]. Moreover, if ν ≥ 2 and there are no loops with non-zero index in the fundamental graph G * , then at least two spectral bands of H are non-degenerate (see Proposition 2.5 in [14]).…”

“…In order to present our results we describe known total bandwidth estimates for Schrödinger operators H = H o +V with periodic potentials V , where the unperturbed operator H o = −∆. There are only upper estimates determined in [14,17,18,19]. We do not know lower estimates.…”

“…Estimates on effective masses for Laplace and Schrödinger operators on periodic graphs were determined in a series of papers [11,16,18]. A band localization for Laplace and Schrödinger operators on periodic graphs was discussed in [6,15,22].…”