Dispersion for the Schrödinger equation on networks

Abstract

In this paper we consider the Schrödinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schrödinger equation with constant coefficients. This allows us to prove dispersive estimates, which in turn are useful for solving the nonlinear Schrödinger equation. The proof extends also to the laminar case of positive step-function coefficients having a finite number of discontinuities.

Soliton transport in tubular networks: Transmission at vertices in the shrinking limit

“…The graph is assumed to have semi-infinite bonds r , = (-o o ,0 ), T2,3 = (0,oo), but the main part of our analysis will be numerical, for which we assume finite lengths lk of bonds, with coordinates ti € (-/ 1,0), ?2,3 € (OJ2. 3), and homogeneous Dirichlet boundary conditions at £1 = -£2,3 = Z 2.3. Furthermore, we assume that the solutions…”

“…Dispersion relations for linear and nonlinear Schrodinger equations on networks are discussed in [3]. The problem of fast solitons on star graphs is treated in [4] where estimates for the transmission and reflection coefficients are obtained in the limit of high velocities.…”

Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications

“…See [4][5][6][7][8][9][10]19]. In [11], Banica and Ignat consider the Schrödinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. They prove dispersive estimates which are useful for solving the linear Schrödinger equation.…”

Stationary nonlinear Schrödinger equation on the graph for the triangle with outgoing bonds

“…[3][4][5][6][7]. Dispersion relations for the linear and NLSE on networks are discussed in [12]. Sine-Gordon equation on metric graphs has been studied in [8,9].…”