The relativistic Kronig-Penney Hamiltonian

William Thomas Payne

Zhu Ya, Qiang Tian

Based on the continuity of the wave function and its derivative, the wave function in Kronig-Penney model is analysed. We obtain a transfer matrix connecting the amplitudes of the transmitted and reflected wave function through a barrier. Then the wave function in Kronig-Penney model is derived and discussed.

Fritz Rohrlich

Bruce H. J. McKellar, G. J. Stephenson

We analyze the barrier problem and the Kronig-Penney model for Dirac particles, paying attention to the Klein paradox and the nature of the potential. The ambiguities associated with the \ensuremath{\delta}-function limit are examined and the physically reasonable way to include \ensuremath{\delta}-function potentials in the Dirac-Kronig-Penney problem is identified.

Marek S. Wartak, Ching-Yao Fong

Giuliano Angelone, Domenico Monaco, Gabriele Peluso

We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.