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

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 $v_0\delta (y)+2v_1{\delta }^{\prime }(y)+w_0\delta (y-q)+2w_1{\delta }^{\prime }(y-q)$, we obtain a new point potential of the type $u_0\delta (y)+2u_1{\delta }^{\prime }(y)$, 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 $S{L}_2(\mathbb{R})$. 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.

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• Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schrödinger operator in one dimension
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• Bound States and Resonance Analysis of One-Dimensional Relativistic Parity-Symmetric Two Point Interactions