Strong and weak semiclassical limits for some rough Hamiltonians

TL;DR

This paper investigates the semiclassical limits of the Schrödinger equation with irregular potentials, exploring different topologies and the implications of non-unique classical flows on the limit behavior.

Contribution

It provides new insights into the semiclassical limits for rough Hamiltonians where classical flow uniqueness is not guaranteed, considering multiple topologies and bicharacteristics.

Findings

Different topologies for the semiclassical limit are analyzed.

The case of non-unique classical flows from the same initial point is examined.

Results highlight the complexity of semiclassical limits with irregular potentials.

Abstract

We present several results concerning the semiclassical limit of the time dependent Schr\"odinger equation with potentials whose regularity doesn't guarantee the uniqueness of the underlying classical flow. Different topologies for the limit are considered and the situation where two bicharateristics can be obtained out of the same initial point is emphasized.