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

Abstract

Abstract. Several families of one-point interactions are derived from the system consisting of two and three δ-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 δ-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter ε → 0, so that the intensity of each δ-potential is c j = a j ε 1−µ , a j ∈ R, j = 1, 2, 3, the width of each layer l = ε and the distance between the layers r = cε τ , 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 < µ < 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 µ to the critical value µ = 2, three types of splitting these sets into countable families of resonance curves and surfaces are observed.

A phenomenon of splitting resonant-tunneling one-point interactions

“…In particular, on the plane Q K we have defined Kurasov's δ ′ K -interaction [7] for which the diagonal element θ in the transmission matrix (2) is given by Eq. (34). Under approaching the limiting sets of this plane, the countable splitting of the δ ′ K -interaction occurs that describes the resonant tunneling through the system.…”

“…we obtain the limit transmission matrix Λ in the form of (2). Under the assumption (34), we obtain Kurasov's δ ′ -interaction with the intensity γ ∈ R \ {±2} defined in the distributional sense on the space of discontinuous at x = 0 test functions. For this case one can find the resonance values of a 1 and a 2 as functions of the strength γ:…”

Point Interactions With Bias Potentials

“…However, this resonant-tunneling behavior contradicts with the Λ-matrix (7) where the element θ continuously depends on strength γ. It is remarkable that this controversy can be resolved using the one-dimensional model for the heterostructure consisting of two or three squeezed parallel homogeneous layers approaching to one point [52,53]. Here a "splitting" effect of one-point interactions has been described.…”

Some Remarks on 1D Schrödinger Operators With Localized Magnetic and Electric Potentials

“…Recently a class of the Schrödinger operators with piece-wise constant δ ′ -potentials were studied by Zolotaryuk a.o. [32][33][34][35]; the resonances in the transmission probability for the scattering problem were established. As was shown in [18][19][20]25] these resonances deal with the existence of zero-energy resonances and the halfbound states for singular localized potentials.…”