Spectral band localization for Schrödinger operators on discrete periodic graphs

Abstract

We consider Schrödinger operators on periodic discrete graphs. It is known that the spectrum of these operators has band structure. We describe a localization of spectral bands and estimate the Lebesgue measure of the spectrum in terms of eigenvalues of Dirichlet and Neumann operators on a fundamental domain of the periodic graph. The proof is based on the Floquet decomposition of Schrödinger operators and the minimax principle.Thus, the intervals J n and J n defined by (1.24), (1.25) and their intersections J n ∩ J n , n ∈ N 3 , have the form (2.38) J 1 = [0; 6], J 1 = [0, 4], σ 1 = [0; 2] ⊂ J 1 ∩ J 1 = J 1 = [0; 4], J 2 = [2; 12], J 2 = [0; 4], σ 2 ≈ [2.5; 4] ⊂ J 2 ∩ J 2 = [2; 4], J 3 ≈ [2.5; 12], J 3 ≈ [6; 9.5], σ 3 ≈ [6; 9.5] = J 3 ∩ J 3 = J 3 .Remark. 1) Theorem 1.1 determines the existence of the second spectral gap (see Figure 3c). The intersection of the intervals J n and J n , n = 1, 2, 3, gives more Licensed to Imperial College London. Prepared on Mon Apr 6 16:39:54 EDT 2015 for download from IP 155.198.30.43/216.158.251.45.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Spectral gaps and discrete magnetic Laplacians

“…Magnetic Laplacians and Laplacians on Abelian covering graphs are discussed also in [HS99,HS04]. Korotyaev and Saburova treated discrete Schrödinger operators on discrete graphs in a series of articles (see, e.g., [KS14,KS15,KS17] and references therein).…”

“…In [KS15] the authors develop a bracketing technique similar to the Dirichlet-Neumann bracketing mentioned before and proved estimates of the position of the spectral bands for the combinatorial Laplacian in terms of suitable Neumann and Dirichlet eigenvalue intervals. Moreover, they give an upper estimate of the total band length in terms of these eigenvalues and some geometric data of the graph; this method is extended in [KS17] to the case of magnetic Laplacians with periodic magnetic vector potentials.…”

Laplacians on periodic graphs with guides

“…Note that the estimate (1.6) also holds true for magnetic Schrödinger operators with periodic magnetic and electric potentials (see [KS17]). Estimates of the Lebesgue measure of the spectrum of H 0 in terms of eigenvalues of Dirichlet and Neumann operators on a fundamental domain of the periodic graph were described in [KS15]. Estimates of effective masses, associated with the ends of each spectral band, in terms of geometric parameters of the graphs were obtained in [KS16].…”

Schrödinger operators with guided potentials on periodic graphs

“…They studied its Bloch variety and its integrated density of states. In [LP08], [KS15] the positions of the spectral bands of the Laplacians were estimated in terms of eigenvalues of the operator on finite graphs (the so-called eigenvalue bracketing). The estimate of the total length of all spectral bands σ n (H 0 ) ν n=1 |σ n (H 0 )| 2β, (1.7)…”

“…Note that the estimate (1.7) also holds true for magnetic Schrödinger operators with periodic magnetic and electric potentials (see [KS17]). Estimates of the Lebesgue measure of the spectrum of H 0 in terms of eigenvalues of Dirichlet and Neumann operators on a fundamental domain of the periodic graph were described in [KS15]. Estimates of effective masses, associated with the ends of each spectral band of the Laplacian, in terms of geometric parameters of the graphs were obtained in [KS16].…”