On the Bloch Theorem and Orthogonality Relations

TL;DR

This paper revisits Bloch theorem for periodic operators, revealing new orthogonality relations and introducing modified Wannier functions in k-space, which could enhance computational methods for periodic structures.

Contribution

It uncovers additional orthogonality relationships and defines invertible modified Wannier functions in k-space, offering potential for new algorithms and applications.

Findings

Discovered extra orthogonality relations for Bloch solutions.

Defined invertible modified Wannier functions in k-space.

Proposed potential for improved computational algorithms.

Abstract

Bloch theorem for a periodic operator is being revisited here, and we notice extra orthogonality relationships. It is shown that solutions are bi-periodic, in the sense that eigenfunctions are periodic with respect to one argument, and pseudo-periodic with respect to the other. An additional kind of symmetry between r-space and k-space exists between the envelope and eignfunctions not apparently noticed before, which allows to define new invertible modified Wannier functions. As opposed to the Wannier functions in r-space, these modified Wannier functions are defined in the k-spaces, but satisfy similar basic properties. These could result in new algorithms and other novel applications in the computational tools of periodic structures.