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Journal of Physics A Mathematical and General

Point interactions in one-dimensional quantum mechanics with coupled channels

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

Published Feb 11, 2004• 4 citations
Publisher

Abstract

We explore generalized point interactions in one-dimensional quantum mechanics with two coupled channels. They represent possible self-adjoint extensions of the nonrelativistic kinetic-energy operator. Assuming time-reversal invariance we find a family of self-adjoint extensions with seven parameters. This can be compared with the one-channel case in which the corresponding number of parameters is three.

DOI
10.1088/0305-4470/37/8/010

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We explore generalized point interactions in one-dimensional quantum mechanics with two coupled channels. They represent possible self-adjoint extensions of the nonrelativistic kinetic-energy operator. Assuming time-reversal invariance we find a family of self-adjoint extensions with seven parameters. This can be compared with the one-channel case in which the corresponding number of parameters is three.
There is a four-parameter family of point interactions in one-dimensional quantum mechanics. They represent all possible self-adjoint extensions of the kinetic energy operator. If time-reversal invariance is imposed, the number of parameters is reduced to three. One of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known. We present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so-called interaction. We emphasize that there is little resemblance between the interaction and what its name suggests.
Wu and Yu recently examined point interactions in one dimension in the form of the Fermi pseudo-potential. On the other hand there are point interactions in the form of self-adjoint extensions (SAEs) of the kinetic energy operator. We examine the relationship between the point interactions in these two forms in the one-channel and two-channel cases. In the one-channel case the pseudo-potential leads to the standard three-parameter family of SAEs. In the two-channel case the pseudo-potential furnishes a ten-parameter family of SAEs.
We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schr\"odinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called δ′\delta^\primeδ′ interaction.
We consider a field theory describing interacting nonrelativistic particles of two types, which map to each other under time reversal, with pointlike interaction. We identify an alternative type of interaction which depends on the relative velocity between the particles. We compute the renormalization-group running of the coupling constants and find a fixed point and a fixed line. We show that the scattering amplitudes can be expressed in terms of three parameters. The result matches with a quantum-mechanical analysis and represents the most general pointlike interaction consistent with unitarity and time-reversal invariance.