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Description
In this work, we consider non-Hermitian Hamiltonians exhibiting $\mathcal{PT}$ symmetry, which possess real energy spectra despite their complex potentials. We study in detail the generalized Kronig-Penney model with a complex periodic potential of the Dirac delta form to explore its electronic properties. By analyzing special cases, we demonstrate the effect of the complex form of the potential on the dispersion relation, the corresponding band structure, the densities of electronic states, and the spatial distributions of the eigenstates. Our analysis shows that already for real potentials of the Kronig-Penney type, which alternately change sign, the band structure exhibits no forbidden energy gaps in the Brillouin zone center. In contrast, for complex potentials of this type, energy gaps never occur at the Brillouin zone edges. Furthermore, analysis of the electronic eigenstates shows that the effect of specific wave-function matching conditions on the Dirac delta varies depending on the nature of the potential. For the real potential, we observe a jump in the derivative of the wave–function modulus, while the derivative of the wavefunction phase is continuous. For the imaginary potential, the jump is observed in the derivative of the wave-function phase, whereas the wave-function modulus remains smooth. In the general complex case, a jump in the derivative of the wave function on the Dirac delta potential is observed for both its modulus and phase. We demonstrate these facts analytically, while using numerical calculations just for illustration.