On Schroedinger Equations with Concentrated Nonlinearities

TL;DR

This paper analyzes Schrödinger equations with localized nonlinearities, demonstrating that in the semiclassical limit, the complex system simplifies to coupled ordinary differential equations and linear Schrödinger equations, aiding understanding of physical phenomena like resonant tunneling.

Contribution

It introduces a method to reduce complex nonlinear Schrödinger equations with localized nonlinearities to simpler coupled equations in the semiclassical limit, providing new insights into their mathematical and physical properties.

Findings

Reduction of nonlinear Schrödinger equations to coupled ODEs and linear equations in the semiclassical limit

Application to resonant tunneling in double barrier heterostructures

Numerical prediction of nonlinear oscillations in the model

Abstract

Schr\"odinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to separate the relevant degrees of freedom by noticing that in the regions where the nonlinearities are effective all states are suppressed but the metastable ones (resonances). In this way the description of the nonlinear regions is reduced to ordinary differential equations weakly coupled to standard Schr\"odinger equations valid in the linear regions. The idea is illustrated through the study of a prototype equation recently proposed for resonant tunneling of electrons through a double barrier heterostructure and for which nonlinear oscillations have been numerically predicted.