Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry

TL;DR

This paper develops a generalized Bloch analysis framework for quantum Hamiltonians on Riemannian manifolds with discrete symmetries, decomposing the space into equivariant functions and relating propagators to boundary conditions.

Contribution

It introduces a novel approach to analyze invariant quantum Hamiltonians on Riemannian manifolds with symmetry groups, linking quasi-periodic boundary conditions to propagator expressions.

Findings

Decomposition of $L^2$ space into equivariant functions.

Expression of propagators in terms of boundary conditions.

Mutual inverse relationship between the procedures.

Abstract

We consider an invariant quantum Hamiltonian $H=-\Delta_{LB}+V$ in the $L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a countable discrete symmetry group $\Gamma$. Typically, $\tilde{M}$ is the universal covering space of a multiply connected Riemannian manifold $M$ and $\Gamma$ is the fundamental group of $M$. On the one hand, following the basic step of the Bloch analysis, one decomposes the $L^{2}$ space over $\tilde{M}$ into a direct integral of Hilbert spaces formed by equivariant functions on $\tilde{M}$. The Hamiltonian $H$ decomposes correspondingly, with each component $H_{\Lambda}$ being defined by a quasi-periodic boundary condition. The quasi-periodic boundary conditions are in turn determined by irreducible unitary representations $\Lambda$ of $\Gamma$. On the other hand, fixing a quasi-periodic boundary condition (i.e., a unitary representation $\Lambda$ of $\Gamma$) one can express the corresponding propagator in terms of the propagator associated to the Hamiltonian $H$. We discuss these procedures in detail and show that in a sense they are mutually inverse.