Semiclassical analysis of the Schr{\"o}dinger equation with conical singularities

TL;DR

This paper investigates how quantum states evolve in the Schrödinger equation with conical singularities, revealing a dual Hamiltonian flow structure and addressing classical trajectory non-uniqueness.

Contribution

It introduces a detailed analysis of Wigner measure propagation involving two Hamiltonian flows and explores the transfer phenomena between regular and singular regions.

Findings

Wigner measures follow two distinct Hamiltonian flows.

Mass concentration occurs around the singular set.

Classical trajectories are not unique at singularities, but quantum solutions remain unique.

Abstract

In this article we study the propagation of Wigner measures linked to solutions of the Schr{\"o}dinger equation with potentials presenting conical singularities and show that they are transported by two different Hamiltonian flows, one over the bundle cotangent to the singular set and the other elsewhere in the phase space, up to a transference phenomenon between these two regimes that may arise whenever trajectories in the outsider flow lead in or out the bundle. We describe in detail either the flow and the mass concentration around and on the singular set and illustrate with examples some issues raised by the lack of unicity for the classical trajectories at the singularities despite the unicity for the quantum solutions, dismissing any classical selection principle, but in some cases being able to fully solve the propagation problem.