Invariants for Laplacians on periodic graphs

Abstract

We consider a magnetic Laplacian with periodic magnetic potentials on periodic discrete graphs. Its spectrum consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We obtain a specific decomposition of the magnetic Laplacian into a direct integral in terms of minimal forms. A minimal form is a periodic function defined on edges of the periodic graph with a minimal support on the period. It is crucial that fiber magnetic Laplacians (matrices) have the minimal number of coefficients depending on the quasimomentum and the minimal number of coefficients depending on the magnetic potential. We show that these numbers are invariants for the magnetic Laplacians on periodic graphs. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the magnetic Laplacian spectrum and a variation of the spectrum under a perturbation by a magnetic field in terms of these invariants and minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber magnetic Laplacians. Moreover, similar results for magnetic Schrödinger operators with periodic potentials are obtained.

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

“…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].…”

Spectral estimates for Schrödinger operators on periodic discrete graphs

“…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 [13,16,17,18]. We do not know lower estimates.…”

“…Then using (4.6), we obtain the lower estimate in (4.12). The upper estimate in (4.12) was proved in [18].…”

“…(a) We recall some properties of the number I (see Theorem 2.1.v and Proposition 2.2 in [18]). ii) The invariant I satisfies…”

“…2) The upper estimate in (2.7) was proved in [18]. The proof of the lower estimate in (2.7) consists of the following steps: • The fiber Schrödinger operators are represented as fiber adjacency operators on a modified quotient graph with additional loops and with specific weights.…”