Point Interactions

Christian Grosche

A proposal is given, of how to implement point interactions and boundary conditions in the path integral. The starting point is a path-integral formulation of a Dirac particle in one dimension. The implementation of a point interaction yields, by means of a perturbation expansion, the corresponding Green function for a relativistic particle. In the non-relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen. By a proper combination of the various possibilities of implementing the point interaction, the whole range of the four-parameter family of boundary conditions for point interactions can be exploited, in the relativistic case as well as in the non-relativistic limit. In addition, making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line. In particular, Dirichlet and Neumann boundary conditions emerge. The method is illustrated with some examples.

Sergio A Albeverio, S.-M. Fei, Pavel Kurasov

General point interactions for the second derivative operator in one dimension are studied. In particular, ${\mathcal P \mathcal T}$-self-adjoint point interactions with the support at the origin and at points $\pm l$ are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.

W.T.M. Caspers

E. de Prunelé

A solvable model for non relativistic quantum scattering in three-dimensional space of a particle off several interactions centred at n arbitrary points is given. The interaction at one centre can be very different from the interaction at another. Each interaction centred at a given point can be effective on any partial wave with respect to this point. Each centred interaction is a sum of projectors on a finite number of partial waves relative to the centre. Elementary applications are given.