Spectroscopy of a one-dimensional V-shaped quantum well with a point impurity

We consider the one-dimensional Hamiltonian with a V-shaped potential H0=12−d2dx2+|x|, decorated with a point impurity of either δ-type, or local δ′-type or even nonlocal δ′-type, thus yielding three exactly solvable models. We analyse the behaviour of the change in the energy levels when an interaction of the type −λδ(x) or −λδ(x−x0) is switched on. In the first case, even energy levels, pertaining to antisymmetric bound states, remain invariant with respect to λ even though odd energy levels, pertaining to symmetric bound states, decrease as λ increases. In the second, all energy levels decrease when the factor λ increases. A similar study has been performed for the so-called nonlocal δ′ interaction, requiring a coupling constant renormalisation, which implies the replacement of the form factor λ by a renormalised form factor β. In terms of β, odd energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of β the energy of each even level, with the natural exception of the first one, becomes lower than the constant energy of the previous odd level. Finally, we consider an interaction of the type −λδ(x)+μδ′(x), and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.