Stationary-state completeness of the Dirac comb

Abstract

We examine the completeness of stationary states of a Hermitian Hamiltonian in the context of the one-dimensional Dirac comb. This system provides a nontrivial realization of the spectral theorem in the presence of energy bands and gaps, where Bloch states replace simple plane waves. Without invoking Fourier theory or rigged Hilbert space machinery, we explicitly construct the stationary states and test their completeness by reconstructing regulated plane waves. By expanding these states over Bloch eigenfunctions across all bands, we demonstrate how contributions from multiple bands conspire to reproduce simple trigonometric functions. This provides a concrete and physically transparent illustration of completeness in a system with band structure.