Quantum particle across Grushin singularity

TL;DR

This paper analyzes quantum particles on Grushin cylinders, revealing how geometry influences transmission, reflection, and confinement, with detailed spectral and scattering analysis of various quantum models across the singularity.

Contribution

It provides a comprehensive spectral and scattering analysis of quantum particles on Grushin cylinders, exploring different transmission/reflection protocols and their spectral properties.

Findings

Identification of non-equivalent transmission/reflection protocols.

Characterization of spectral properties including ground states.

Calculation of transmission and reflection coefficients.

Abstract

A class of models is considered for a quantum particle constrained on degenerate Riemannian manifolds known as Grushin cylinders, and moving freely subject only to the underlying geometry: the corresponding spectral analysis is developed in detail in view of the phenomenon of transmission across the singularity that separates the two half-cylinders. Whereas the classical counterpart always consists of a particle falling in finite time along the geodesics onto the metric's singularity locus, the quantum models may display geometric confinement, or on the opposite partial transmission and reflection. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection and the structure of their spectrum is characterised, including when applicable their ground state and positivity. Besides, the stationary scattering analysis is developed and transmission and reflection coefficients are calculated. This allows to comprehend the distinguished status of the so-called `bridging' transmission protocol previously identified in the literature, which we recover and study within our systematic analysis.