Today: Geometric Structures seminar - Lucia Tessarolo (Paris) - Room 134 - 16.00

[Luca Rizzi]

Dear All,

This is to remind you of today's seminar from the series "Geometric Structures". Note the unusual time: 16:00, room 134

Speaker: Lucia Tessarolo

Title: Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds

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 talk we analyse the Schrödinger evolution of a particle constrained on $\mathscr{F}$.
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. We then discuss self-adjoint extensions of an operator defined on the whole foliation and we construct a special family of such extension in a toy model.

More information can be found here: https://researchseminars.org/seminar/Geometric_Structures_SISSA

​​Luca Rizzi
SISSA (Trieste)
https://rizzilu.perso.math.cnrs.fr/