Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal ${\delta }^{\prime }$ interactions

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 ${\delta }^{\prime }$-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 ${\delta }^{\prime }$ interactions vanishes.

24 pages, 20 figures

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• Self-interacting quantum particles and the Dirac delta potential
• Quantum self-interaction within an infinitely deep cavity
• Bound States and Resonance Analysis of One-Dimensional Relativistic Parity-Symmetric Two Point Interactions
• Generalized Neumann boundary condition for the scalar field