Schr\"odinger equation with singular position dependent mass

TL;DR

This paper studies the Schr"odinger equation with highly singular position-dependent mass, establishing its well-posedness, uniqueness, and consistency with classical theory, enabling analysis of delta-like mass distributions.

Contribution

It introduces a framework for analyzing Schr"odinger equations with singular mass distributions, including delta-like cases, and proves key theoretical properties.

Findings

Proves very weak well-posedness of the equation.

Establishes uniqueness of solutions.

Demonstrates consistency with classical Schr"odinger theory.

Abstract

We consider the Schr\"odinger equation with singular position dependent effective mass and prove that it is very weakly well posed. A uniqueness result is proved in an appropriate sense, moreover, we prove the consistency with the classical theory. In particular, this allows one to consider Delta-like or more singular masses.