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.

arXiv comment24 pages, 20 figures

References in corpus (23)

• Attractive and Repulsive Casimir Vacuum Energy with General Boundary Conditions
• Potential approximations to ${\delta }^{\prime }$: an inverse Klauder phenomenon with norm-resolvent convergence
• Quantum vacuum energies and Casimir forces between partially transparent $\delta$-function plates
• Casimir Effect and Global Theory of Boundary Conditions
• Bose-Einstein condensate in a harmonic trap decorated with Dirac delta functions
• Exact solution of a 1D many-body system with momentum dependent interactions
• One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials
• Double-delta potentials: one dimensional scattering. The Casimir effect and kink fluctuations
• Asymptotic Freedom, Dimensional Transmutation, and an Infra-red Conformal Fixed Point for the $\delta$-Function Potential in 1-dimensional Relativistic Quantum Mechanics
• QFT over the finite line. Heat kernel coefficients, spectral zeta functions and selfadjoint extensions
• Physical interpretation of point-like interactions of one-dimensional Schrödinger operator
• Monoatomically thin polarizable sheets
• Quantum vacuum interaction between two sine-Gordon kinks
• Spectral properties of a short-range impurity in a quantum dot
• Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials
• Nonlinear Schroedinger equation with two symmetric point interactions in one dimension
• The hyperbolic step potential: Antibound states, SUSY partners and Wigner time delays
• Band spectra of periodic hybrid $\delta$-${\delta }^{\prime }$ structures
• The Solution of the Relativistic Schrodinger Equation for the ${\delta }^{\prime }$-Function Potential in 1-dimension Using Cutoff Regularization
• Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schrödinger operator in one dimension
• Illustrations of loosely-bound and resonant states in atomic nuclei
• On the Heisenberg condition in the presence of redundant poles of the S-matrix
• Rank One Perturbations Supported by Hybrid Geometries and Their Deformations

Cited by in corpus (5)

• On negative eigenvalues of 1D Schrödinger operators with ${\delta }^{\prime }$-like potentials
• 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