Schr\"odinger evolution on surfaces in 3D contact sub-Riemannian manifolds

TL;DR

This paper investigates the Schr"odinger evolution constrained to characteristic foliations on surfaces within 3D contact sub-Riemannian manifolds, linking self-adjointness of the operator to geometric invariants and classifying special extensions.

Contribution

It introduces a geometric perspective on Schr"odinger operators on characteristic foliations and classifies self-adjoint extensions with disjoint dynamics in this setting.

Findings

Self-adjointness relates to a geometric invariant of the foliation.

Classification of self-adjoint extensions with disjoint dynamics.

Analysis of Schr"odinger evolution constrained to characteristic foliations.

Abstract

Let $M$ be a 3-dimensional contact sub-Riemannian manifold and $S$ a surface embedded in $M$. Such a surface inherits a field of directions that becomes singular at characteristic points. The integral curves of such field define a characteristic foliation $\mathscr{F}$. In this paper we study the Schr\"odinger evolution of a particle constrained on $\mathscr{F}$. In particular, we relate the self-adjointness of the Schr\"odinger operator with a geometric invariant of the foliation. We then classify a special family of its self-adjoint extensions: those that yield disjoint dynamics.