Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Abstract

Several families of one-point interactions are derived from the system consisting of two and three $\delta$-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of $\delta$-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter $\varepsilon \to 0$, so that the intensity of each $\delta$-potential is $c_j =a_j \varepsilon^{1-\mu}$, $a_j \in {\mathbb{R}}$, $j=1,2,3$, the width of each layer $l =\varepsilon$ and the distance between the layers $r = c\varepsilon^\tau$, $c >0$. It is shown that at some values of intensities $a_1$, $a_2$ and $a_3$, 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. Within the interval $1 < \mu < 2$, the resonance sets consist of two curves on the $(a_1,a_2)$-plane and three disconnected surfaces in the $(a_1,a_2,a_3)$-space. While approaching the parameter $\mu $ to the critical value $\mu =2$, three types of splitting these sets into countable families of resonance curves and surfaces are observed.