Transmission resonances in scattering by $\delta'$-like combs

Abstract

We introduce a new exactly solvable model in quantum mechanics that describes the propagation of particles through a potential field created by regularly spaced $\delta'$-type point interactions, which model the localized dipoles often observed in crystal structures. We refer to the corresponding potentials as $\delta'_\theta$-combs, where the parameter $\theta$ represents the contrast of the resonant wave at zero energy and determines the interface conditions in the Hamiltonians. We explicitly calculate the scattering matrix for these systems and prove that the transmission probability exhibits sharp resonance peaks while rapidly decaying at other frequencies. Consequently, Hamiltonians with $\delta'_\theta$-comb potentials act as quantum filters, permitting tunnelling only for specific wave frequencies. Furthermore, for each $\theta > 0$, we construct a family of regularized Hamiltonians approximating the ideal model and prove that their transmission probabilities have a similar structure, thereby confirming the physical realizability of the band-pass filtering effect.