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Journal of Physics A Mathematical and General

Energy-dependent point interactions in one dimension

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, Lauro Tomio, F. Masafumi Toyama

Published May 18, 2005• 12 citations
Publisher

Abstract

We consider a new type of point interaction in one-dimensional quantum mechanics. It is characterized by a boundary condition at the origin that involves the second and/or higher order derivatives of the wavefunction. The interaction is effectively energy dependent. It leads to a unitary S-matrix for the transmission-reflection problem. The energy dependence of the interaction can be chosen such that any given unitary S-matrix (or the transmission and reflection coefficients) can be reproduced at all energies. Generalization of the results to coupled-channel cases is discussed.

DOI
10.1088/0305-4470/38/22/020

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We consider a new type of point interaction in one-dimensional quantum mechanics. It is characterized by a boundary condition at the origin that involves the second and/or higher order derivatives of the wavefunction. The interaction is effectively energy dependent. It leads to a unitary S-matrix for the transmission-reflection problem. The energy dependence of the interaction can be chosen such that any given unitary S-matrix (or the transmission and reflection coefficients) can be reproduced at all energies. Generalization of the results to coupled-channel cases is discussed.
Strongly singular potentials in one dimension are analyzed, namely the n th derivative interaction     x n  .For the repulsive case, the reflection and transmission coefficients are derived.It is shown that for n = even, these coefficients satisfy the unitarity of the scattering matrix for all values of the incident energy and the barrier becomes perfectly reflective for a specific value of the energy.For n=odd 1  , however, the coefficients do not satisfy the unitarity of the scattering matrix except for one specific value of the incident energy at which the barrier becomes perfectly reflective.For the n=1 case, the results showed that the coefficients satisfy the unitarity of the scattering matrix.For the attractive case, both the bound states and the scattering states are examined.It is shown that, for n=even 0  , there exist two bound states and only one bound state for n=0.For the scattering states, it is demonstrated that one recovers the same coefficients that were obtained for the repulsive case.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
In this paper we argue that boundary condition may run with energy scale. As an illustrative example, we consider one-dimensional quantum mechanics for a spinless particle that freely propagates in the bulk yet interacts only at the origin. In this setting we find the renormalization group flow of U(2) family of boundary conditions exactly. We show that the well-known scale-independent subfamily of boundary conditions are realized as fixed points. We also discuss the duality between two distinct boundary conditions from the renormalization group point of view. Generalizations to conformal mechanics and quantum graph are also discussed.
For a pair of non-Hermitian Hamiltonian H and its Hermitian adjoint H(dagger), there are situations in which their eigenfunctions form a biorthogonal system. We illustrate such a situation by means of a one-particle system with a one-dimensional point interaction in the form of the Fermi pseudo-potential. The interaction consists of three terms with three strength parameters g(i) (i = 1, 2 and 3), which are all complex. This complex point interaction is neither Hermitian nor PT-invariant in general. The S-matrix for the transmission reflection problem constructed with H (or with H(dagger)) in the usual manner is not unitary, but it conforms to the pseudo-unitarity that we define. The pseudounitarity is closely related to the biorthogonality of the eigenfunctions. The eigenvalue spectrum of H with the complex interaction is generally complex but there are cases where the spectrum is real. In such a case H and H(dagger) form a pseudo-Hermitian pair.