Initial value problem for the time-dependent linear Schr\"odinger equation with a point singular potential by the uniform transform method

TL;DR

This paper applies the Fokas unified transform method to solve the initial value problem for the linear Schrödinger equation with a point singular potential, providing integral representations of solutions.

Contribution

It extends the unified transform method to handle Schrödinger equations with point singular potentials, offering a new approach for such problems.

Findings

Derived integral representation of the solution

Reformulated the problem as coupled IBV problems on two half-lines

Paves the way for solving nonlinear Schrödinger equations with singular potentials

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.