On the Bloch Theorem and Orthogonality Relations

Sina Khorasani

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arXiv · 1505.00319

On the Bloch Theorem and Orthogonality Relations

Sina Khorasani

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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.

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Sina Khorasani. 2015-05-02. On the Bloch Theorem and Orthogonality Relations. https://arxiv.org/abs/1505.00319

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