Relativistic Point Interactions Research Articles

The physical interpretation of point interactions in one-dimensional relativistic quantum mechanics

We investigate point interactions (PIs) in one-dimensional relativistic quantum mechanics using a distributional approach based on Schwartz’s theory of distributions. From the properties of the most general covariant distribution describing relativistic PIs (RPIs) we obtain the physical parameters associated with the point potentials that behave as a scalar, a pseudo-scalar and a vector under Lorentz transformations. Then, we establish a one-to-one relationship between these physical parameters and the well-known set of four parameters giving the boundary conditions at the singular point(s), which define a self-adjoint Hamiltonian. By considering the non-relativistic limit, we obtain the most general PI in the Schrödinger equation in terms of these four physical point potentials. Finally, we study the symmetries of the RPIs under space inversion, time reversal and charge conjugation, and investigate how requirements of invariance under these symmetry transformations can be used to restrict the set of physical parameters.

Non-self-adjoint relativistic point interaction in one dimension

The one-dimensional Dirac operator with a singular interaction term which is formally given by A⊗|δ0〉〈δ0|, where A is an arbitrary 2×2 matrix and δ0 stands for the Dirac distribution, is introduced as a closed not necessarily self-adjoint operator. We study its spectral properties, find its non-relativistic limit and also address the question of regular approximations. In particular, we show that, contrary to the case of local approximations, for non-local approximating potentials, coupling constants are not renormalized in the limit.

Approximation of one-dimensional relativistic point interactions by regular potentials revised

We show that the one-dimensional Dirac operator with quite general point interaction may be approximated in the norm resolvent sense by the Dirac operator with a scaled regular potential of the form $1/\varepsilon~h(x/\varepsilon)\otimes B$, where $B$ is a suitable $2\times 2$ matrix. Moreover, we prove that the limit does not depend on the particular choice of $h$ as long as it integrates to a constant value.

Smooth approximation of finitely many relativistic point interactions

The purpose of this paper is to show that finitely many relativistic point interactions may be approximated in the strong resolvent sense by sequences of operators with smooth potentials. We consider the entire family of relativistic point interactions, and determine those for which renormalization of the coupling constant occurs when the corresponding potentials are approximated by finitely many local, short-range perturbations of the free Dirac operator.

Finite-Rank Perturbations of the Dirac Operator

In this paper we determine those relativistic point interactions for which renormalization of the coupling constant occurs when the corresponding potentials are approximated by local, short-range perturbations of the free Dirac operator in one dimension. We also find a general formula for the renormalization constant for the entire four-parameter family of relativistic point interactions. Our results, which unify and extend earlier work on relativistic point interactions, include perturbations of the Dirac operator by finitely many δ or δ′ potentials, as well as relativistic scalar and magnetic point interactions. In addition, we show that similar methods may be used to define more general perturbations of the Dirac operator, including time-dependent ones. These operators are shown to depend continuously on the perturbation in the sense of strong resolvent convergence.

Relativistic point interactions: Approximation by smooth potentials

We show that the four-parameter family of one-dimensional relativistic point interactions studied by Benvegnù and Da̧browski may be approximated in the strong resolvent sense by smooth, local, short-range perturbations of the Dirac Hamiltonian. In addition, we prove that the nonrelativistic limits correspond to the Schrödinger point interactions studied extensively by the author and Paul Chernoff.

Relativistic point interaction with Coulomb potential in one dimension

The Dirac Hamiltonian in one space dimension is investigated under the influence of a potential of the form −γ/|x|. The corresponding (four-parameter) family of all self-adjoint extensions is given and described via the boundary form. The resolvent is calculated and the spectrum is studied. Furthermore, we examine the zero mass case. In the nonrelativistic limit we obtain the four-parameter family of Schrödinger operators with the Coulomb potential.

Delta '-function perturbations and Neumann boundary conditions by path integration

$\delta'$-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual $\delta$-function or a $\delta'$-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral.

Scattering states of relativistic point interaction potentials

Scattering of positive and negative energy Dirac particles moving under the action of vector and scalar point interaction potentials faced to an impenetrable wall has been discussed. The occurrence of a well defined resonance pattern is only found for scalar couplings.

Klein's paradox and the relativistic point interaction

On considere une approximation a courte portee pour des potentiels relativistes en fonction delta

Bound states and confining properties of relativistic point interaction potentials

Bound states of the one-dimensional Dirac equation for vector plus Lorentz scalar point interaction potentials have been obtained. Confining properties of these potentials are briefly discussed.

New analytically solvable models of relativistic point interactions

Two new analytically solvable models of relativistic point interactions in one dimension (being natural extensions of the nonrelativistic δ, resp. δ′, interactions) are considered. Their spectral properties in the case of finitely many point interactions as well as in the periodic case are fully analyzed. Moreover, we explicitly determine the spectrum in the case of independent, identically distributed random coupling constants and derive the analog of the Saxon and Hutner conjecture concerning gaps in the energy spectrum of such systems.