Band spectra of periodic hybrid $\delta$-${\delta }^{\prime }$ structures

Abstract

We present a detailed study of a generalised one-dimensional Kronig-Penney model using $\delta \text{-}{\delta }^{\prime }$ potentials. We analyse the band structure and the density of states in two situations. In the first case we consider an infinite array formed by identical $\delta \text{-}{\delta }^{\prime }$ potentials standing at the linear lattice nodes. This case will be known throughout the paper as the one-species hybrid Dirac comb. We investigate the consequences of adding the ${\delta }^{\prime }$ interaction to the Dirac comb by comparing the band spectra and the density of states of pure Dirac-$\delta$ combs and one-species hybrid Dirac combs. Secondly we study the quantum system that arises when the periodic potential is the one obtained from the superposition of two one-species hybrid Dirac combs displaced one with respect to the other and with different couplings. The latter will be known as the two-species hybrid Dirac comb. One of the most remarkable results is the appearance of a curvature change in the band spectrum when the ${\delta }^{\prime }$ couplings are above a critical value.

25 pages, 10 figures. Mayor changes have been done in the structure of the paper

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• Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal ${\delta }^{\prime }$ interactions
• Free energy and entropy for finite temperature quantum field theory under the influence of periodic backgrounds
• Application of a quantum wave impedance method for study of infinite and semi-infinite periodic media
• Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian
• Isospectrality and non-locality of generalized Dirac combs
• Quantum field theory at finite temperature for 3D periodic backgrounds