Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method

Abstract

We study an initial value problem for the one-dimensional non-stationary linear Schr\"odinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem. Since the unified approach is known as providing efficient solutions to both linear and nonlinear problems, the present paper can be viewed as a step in solving the initial value problem for the non-stationary {\em nonlinear} Schr\"odinger equation with a point singular potential.