“On the Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds”
Lunedì 10 Febbraio 2025, ore 17:00 - Aula 1BC50 - Lucia Tessarolo (Phd Student Sorbonne University)
Abstract
Let $M$ be a 3D contact sub-Riemannian manifold and $S$ a surface embedded in $M$. We study the Schrödinger evolution of a particle constrained on the characteristic foliation $F$ of $S$. Specifically, we define the Schrödinger operator $\Delta\ell$ on each leaf $\ell$ as the classical “divergence of gradient”, where the gradient is the Euclidean gradient along the leaf and the divergence is taken with respect to the surface measure inherited from the Popp volume, using the sub-Riemannian normal to the surface. We then study the self-adjointness of the operator $\Delta\ell$ on each leaf by defining a notion of “essential self-adjointness at a point”, in such a way that $\Delta\ell$ will be essentially self-adjoint on the whole leaf if and only if it is essentially self-adjoint at both its endpoints. We see how this local property at a characteristic point depends on a curvature-like invariant at that point. Additionally, we study self-adjoint extensions for the particular case of an infinite leaf.
