Furcation of resonance sets for one-point interactions

Abstract

Families of one-point interactions are derived from the system consisting of regularized two- and three-delta potentials using different paths of the convergence of corresponding transmission matrices in the squeezing limit. This limit is controlled by the relative rate of shrinking the width of delta-like functions and the distance between these functions using the power parameterization: width $l =\varepsilon^{\mu -1}$, $\mu \in [2,\, \infty]$ (for width) and $r = \varepsilon^\tau$, $\tau \in [1,\, \infty]$ (for distance). It is shown that at some values of real coefficients (intensities $a_1$, $a_2$ and $a_3$) at the delta potentials, the transmission across the limit point interactions is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. The resonance sets of intensities at which a non-zero transmission occurs are proved to be of four types depending on the way of squeezing the regularized system to one point. In its turn, on these sets the limit one-point interactions are observed to be either single- or multiple-resonant-tunnelling potentials also depending on the squeezing way. In the two-delta case the resonance sets are curves on the $(a_1,a_2)$-plane and surfaces in the $(a_1,a_2,a_3)$-space for the three-delta system. A new phenomenon of furcation of single-valued resonance sets to multi-valued ones is observed under approaching the parameter $\mu >2$ to the value $\mu =2$.