Title: 

Observability and Semiclassical Control for Schrödinger Equations on Non-compact Covering Spaces of Compact Hyperbolic Surfaces

Speaker: Yulin Gong 龚禹霖 (University of Bristol, UK)

Lecture Room: 5205

Time: 10:00-11:00, 2025.12.22

Abstract:

In this talk, we study the observability of the Schrödinger equation on $X$, a non-compact covering space of a compact hyperbolic surface $M$. Using a generalized Bloch theory, wave functions on $X$ are identified as sections of a unitary flat Hilbert bundle over $M$. We extend the semiclassical analysis to unitary flat Hilbert bundles and generalize Dyatlov and Jin's semiclassical control to all flat unitary Hilbert bundles over $M$, with uniform constants independent of the choice of bundle. Furthermore, if the Riemannian cover $X \to M$ is a normal cover with a virtually Abelian deck transform group $\Gamma$, we apply the generalized Bloch theorem to derive the observability from all $\Gamma$-periodic open subsets of $X$. We will also discuss the application of uniform semiclassical control in spectral geometry. This is joint work with Xin Fu(Tsinghua University) and Yunlei Wang(Louisiana State University).