Quantum ergodicity for restrictions to hypersurfaces

TL;DR

This paper demonstrates that under certain dynamical conditions, the restrictions of eigenstates to a hypersurface in position space exhibit quantum ergodicity, extending the classical quantum ergodicity results.

Contribution

It establishes quantum ergodicity for eigenstate restrictions to hypersurfaces under specific dynamical conditions, a novel extension of existing quantum ergodicity theorems.

Findings

Restrictions of eigenstates to hypersurfaces are quantum ergodic.

The result applies under a simple dynamical condition on the hypersurface.

Extends quantum ergodicity to a new class of geometric restrictions.

Abstract

Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, in average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in the position space satisfying a simple dynamical condition, the restrictions of eigenstates to N are also quantum ergodic.