Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph

Abstract

Abstract. In this paper, we consider periodic Schrödinger operators on the dumbbell-like metric graph, which is a periodic graph consisting of lines and rings. Let one line and two rings be in the basic period. We see the relationship between the structure of graph and the band-gap spectrum.

Validity of the NLS approximation for periodic quantum graphs

“…The spectral problems associated with the linear Schrödinger operator on the periodic quantum graph of Fig. 1 and its modifications have been recently studied in the literature [15][16][17]. Our work is different in the sense that we are studying the time evolution (Cauchy) problem for the nonlinear version of the Schrödinger equation associated with localized initial data.…”

Spectral Band Structure of Periodic Schrödinger Operators on a Generalized Degenerate Zigzag Nanotube

“…We compare our results with the classical results and related results [11,16]. As stated above, the spectrum of H 0 has the band structure and the band edges {λ ± 0,n } satisfy the in-…”

“…We give more precise definition in the next paragraph. Although we later describe the relationship between this paper and [11,16] in detail, we now introduce earlier paper [2,3,5,11,13,18] on the subject of periodic Schrödinger operators on metric graphs in brief before we define our operators. One example of quasi-1-dimensional metric graph is a homogeneous tree, whose vertices have a common number of edges.…”

Bifurcations of Standing Localized Waves on Periodic Graphs

“…on the periodic graph Γ shown in Figure 1. The same periodic graph and its modifications was considered in the previous literature within the linear spectral theory of the associated stationary Schrödinger operator [16,17,20]. In the recent work [11], using the Floquet-Bloch spectral transform and energy methods, we have addressed the time evolution problem associated with the NLS equation (1) on the periodic graph Γ and justified the most universal approximation of the modulated wave packets given by the homogeneous NLS equation…”