Quantization and isotropic submanifolds

TL;DR

This paper introduces isotropic quantum states linked to Bohr-Sommerfeld manifolds within Berezin-Toeplitz quantization, exploring their semi-classical behavior and extending results to orbifolds with applications in automorphic forms.

Contribution

It defines isotropic quantum states in a new geometric setting and analyzes their semi-classical properties, extending the theory to non-compact orbifolds and automorphic forms.

Findings

Characterization of isotropic quantum states in Berezin-Toeplitz quantization

Extension of semi-classical analysis to orbifolds

Application to automorphic forms via Poincaré series

Abstract

We introduce the notion of an isotropic quantum state associated with a Bohr-Sommerfeld manifold in the context of Berezin-Toeplitz quantization of general prequantized symplectic manifolds, and we study its semi-classical properties using the off-diagonal expansion of the Bergman kernel. We then show how these results extend to the case of non-compact orbifolds, and give an application to relative Poincar\'e series in the theory of automorphic forms.