Two-point one-dimensional $\delta$-$\delta^\prime$ interactions: non-abelian addition law and decoupling limit

TL;DR

This paper explores the composition and limits of one-dimensional delta and delta-prime point interactions, revealing a non-abelian addition law, a group structure, and detailed spectral analysis, including boundary condition equivalences.

Contribution

It demonstrates that the limit of combined point potentials forms a new potential with a group structure, and derives a non-abelian addition law from scattering data, with detailed spectral and boundary condition analysis.

Findings

Limit of combined potentials forms a new potential with group structure.

Derived a non-abelian addition law from scattering data.

Identified boundary condition equivalences at specific coupling values.

Abstract

In this contribution to the study of one dimensional point potentials, we prove that if we take the limit $q\to 0$ on a potential of the type $v_0\delta({y})+{2}v_1\delta'({y})+w_0\delta({y}-q)+ {2} w_1\delta'({y}-q)$, we obtain a new point potential of the type ${u_0} \delta({y})+{2 u_1} \delta'({y})$, when $ u_0$ and $ u_1$ are related to $v_0$, $v_1$, $w_0$ and $w_1$ by a law having the structure of a group. This is the Borel subgroup of $SL_2({\mathbb R})$. We also obtain the non-abelian addition law from the scattering data. The spectra of the Hamiltonian in the exceptional cases emerging in the study are also described in full detail. It is shown that for the $v_1=\pm 1$, $w_1=\pm 1$ values of the $\delta^\prime$ couplings the singular Kurasov matrices become equivalent to Dirichlet at one side of the point interaction and Robin boundary conditions at the other side.