Theory of finite periodic systems: The eigenfunctions symmetries

TL;DR

This paper derives symmetry relations of eigenfunctions in finite periodic systems, linking their properties to Chebyshev polynomial zeros and system parameters, enhancing understanding of their quantum behavior.

Contribution

It provides analytical expressions for eigenfunction symmetries in finite periodic systems, connecting these symmetries to Chebyshev polynomial properties and system indices.

Findings

Eigenfunctions exhibit specific space-inversion symmetries related to Chebyshev polynomial zeros.

Symmetry relations depend on the number of unit cells, subband, and intra-subband indices.

Analytical expressions clarify the structure of eigenfunctions in finite periodic systems.

Abstract

Using the analytical expressions for the genuine eigenfunctions $\varphi_{\mu\nu}(z)$ and eigenvalues $E_{\mu,\nu}$, of open, bounded and quasi-bounded finite periodic systems, we derive the eigenfunctions space-inversion symmetry relations. The superlattice eigenfunctions symmetries, closely related with the symmetries and zeros of the Chebyshev polynomials of the second kind $U_n$, are fully written in terms of the number of unit cells $n$, the subband index $\mu$ and the intra-subband index $\nu$.