Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal $\delta'$ interactions

TL;DR

This paper rigorously analyzes a one-dimensional quantum Hamiltonian with two symmetric nonlocal delta prime interactions, exploring its spectrum, resonances, and behavior as the interaction distance approaches zero.

Contribution

It introduces a self-adjoint formulation of a Hamiltonian with nonlocal delta prime interactions and studies its spectral properties and limiting behavior.

Findings

The Hamiltonian has two negative eigenvalues representing bound states.

The energy levels depend on interaction strength and distance between interactions.

The model's behavior as the interaction distance vanishes is thoroughly characterized.

Abstract

The objective of the present paper is the study of a one-dimensional Hamiltonian with the interaction term given by the sum of two nonlocal attractive $\delta'$-interactions of equal strength and symmetrically located with respect to the origin. We use the procedure known as {\it renormalisation of the coupling constant} in order to rigorously achieve a self-adjoint determination for this Hamiltonian. This model depends on two parameters, the interaction strength and the distance between the centre of each interaction and the origin. Once we have the self-adjoint determination, we obtain its discrete spectrum showing that it consists of two negative eigenvalues representing the energy levels. We analyse the dependence of these energy levels on the above-mentioned parameters. We investigate the possible resonances of the model. Furthermore, we analyse in detail the limit of our model as the distance between the supports of the two $\delta'$ interactions vanishes.