Strong Approximation of Stochastic Semiclassical Schroedinger Equation with Multiplicative Noise

TL;DR

This paper develops a numerical method for the stochastic nonlinear Schrödinger equation in the semiclassical regime, accurately capturing high-frequency oscillations and providing convergence rates dependent on the Planck constant.

Contribution

It introduces a spectral Galerkin and midpoint scheme combination with proven strong convergence rates tailored for the semiclassical stochastic Schrödinger equation.

Findings

The method accurately approximates physical observables.

Convergence rate explicitly depends on the Planck constant.

Provides meshing strategies for correct observable computation.

Abstract

We consider the stochastic nonlinear Schroedinger equation driven by a multiplicative noise in a semiclassical regime, where the Plank constant is small. In this regime, the solution of the equation exhibits high-frequency oscillations. We design an efficient numerical method combining the spectral Galerkin approximation and the midpoint scheme. This accurately approximates the solution, or at least of the associated physical observables. Furthermore, the strong convergence rate for the proposed scheme is derived, which explicitly depends on the Planck constant. This conclusion implies the semiclassical regime's admissible meshing strategies for obtaining "correct" physical observables.