Quantum particle across Grushin singularity

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.

Uniform, localized asymptotics for sub-Riemannian heat kernels and diffusions

“…As already noted, we can find δ such that the Ben Arous expansion holds for x, y ∈ K with d(x, y) ≤ δ. Thus the estimates for log p t (x, y) when d(x, y) ≤ δ follow from (25) and compactness. Then the estimates for d(x, y) > δ come from the previous theorem.…”

“…Thus this is not a smooth sub-Laplacian, and indeed, the Léandre asymptotics fail dramatically. For recent work in this direction on Grushin and related structures, see [21,20,26,25].…”