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

摘要

We consider the one-dimensional Hamiltonian with a V-shaped potential H-0 = 1/2 [- d(2)/dx(2) + vertical bar x vertical bar], decorated with a point impurity of either delta-type, or local 3'-type or even nonlocal delta'-type, thus yielding three exactly solvable models. We analyse the behaviour of the change in the energy levels when an interaction of the type -lambda delta(x) or -lambda delta(x - x(0)) is switched on. In the first case, even energy levels, pertaining to antisymmetric bound states, remain invariant with respect to lambda even though odd energy levels, pertaining to symmetric bound states, decrease as lambda increases. In the second, all energy levels decrease when the factor lambda increases. A similar study has been performed for the so-called nonlocal delta' interaction, requiring a coupling constant renormalisation, which implies the replacement of the form factor lambda. by a renormalised form factor beta. In terms of beta, odd energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of beta 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 -lambda delta(x)+mu delta'(x), and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian. (C) 2017 Elsevier Inc. All rights reserved.