Two-point one-dimensional δ-δ’ 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 ${v}_{0}\delta (y)+2{v}_{1}{\delta }^{\prime }(y)+{w}_{0}\delta (y-q)+2{w}_{1}{\delta }^{\prime }(y-q),$ we obtain a new point potential of the type ${u}_{0}\delta (y)+2{u}_{1}{\delta }^{\prime }(y),$ when u0 and u1 are related to v0, v1, w0 and w1 by a law with 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 decoupling cases emerging in the study are also described in full. 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.