Nonlinear Schroedinger equation with two symmetric point interactions in one dimension

TL;DR

This paper studies a one-dimensional nonlinear Schrödinger equation with a symmetric double delta potential, providing explicit formulas, estimates, and proving local existence and uniqueness of solutions.

Contribution

It introduces explicit formulas for the linear propagator and establishes Strichartz estimates for the nonlinear problem with double delta interactions.

Findings

Explicit integral kernel formula for the linear propagator

Strichartz-type estimates for the linear evolution

Local existence and uniqueness of solutions to the nonlinear equation

Abstract

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem.