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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• Band spectra of periodic hybrid $\delta$-${\delta }^{\prime }$ structures
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• The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity
• Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schrödinger operator in one dimension
• Resonant Transmission in One-Dimensional Quantum Mechanics with Two Independent Point Interactions: Full Parameter Analysis
• Scattering data and bound states of a squeezed double-layer structure
• Origin of resonant tunneling through single-point barriers
• Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian
• Regularization of a strong-weak duality between pointlike interactions in one dimension
• Exact results for the Wigner transform phase space densities of a two-dimensional harmonically confined charged quantum gas subjected to a magnetic field
• Exact results for nonequilibrium dynamics in Wigner phase space
• Bound States and Resonance Analysis of One-Dimensional Relativistic Parity-Symmetric Two Point Interactions
• Conditions for realizing one-point interactions from a multi-layer structure model