Perturbation theory for metastable states of the Dirac equation with quadratic vector interaction

Abstract

The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The problem is singular and the perturbation series is asymptotic, so that the methods for dealing with divergent series must be used. Among these, the Distributional Borel Sum appears to be the most well suited tool to give answers and to describe the spectral properties of the system. A detailed investigation is made in one and in three space dimensions with a central potential. We present numerical results for the Dirac equation in one space dimension: these are obtained by determining the perturbation expansion and using the Padé approximants for calculating the distributional Borel transform. A complete agreement is found with previous non-perturbative results obtained by the numerical solution of the singular boundary value problem and the determination of the density of the states from the continuous spectrum.

PT-symmetric operators and metastable states of the 1D relativistic oscillators

“…The largest value of the latter has been taken to be 300: although a large number of decimal figures is already stabilized by few iterations, a very high precision is however necessary for a comparison of the t with the d eigenvalues, that we show here below. As expected, the results obtained in [26,27] when studying the resonances of the Dirac equation by the spectral concentration and by the DB sum, as well as the asymptotic behavior of the imaginary part, are confirmed by the much more precise data given in Table 1. In the application of the Rayleigh-Ritz method we have used as variational parameters the size of the imaginary translation y introduced in item (3.3) and the the 'frequency' σ of the operator H 0 .…”

“…Remark 4.1 This means that each energy level is given by the distributional Borel sum of the perturbation series modulo a correction of the second order on the pair production effect (see [27]).…”

“…Due to the very cumbersome analytical calculations involved, the treatment was essentially restricted to the first perturbation order giving, however, some more explicit information on the properties of the continuous spectrum by investigating the Weyl function m(λ) of the singular boundary value problem [26] in the complex plane. Very recently, numerical investigations of the Dirac equation with a linear and a quadratic electrostatic potential were presented in [27,28], looking for a dissipative model: metastable states were found and the Schwinger pair production rate [29] was calculated in terms of the spectral concentration and in terms of the imaginary part of the resonances of the Dirac equation in external linear and quadratic potentials. In [27] the density of the states [30] was determined at finite values of (1/c), finding a sum of Breit-Wigner lines whose width reproduced the pair production rate.…”

Two body relativistic wave equations

“…In a paper of the same year [9] Plesset proved a result contrary to a naif intuition based on a non-relativistic picture: namely he showed that asymptotically unbounded positive potentials in vector coupling have no bound states. More detailed descriptions of the continuous spectrum for such and similar situations have been given in [10][11][12]. In fact the covariance properties were not the first concern of those papers and the equations assumed explicitly the reference frame with vanishing total momentum.…”

Klein tunneling and Dirac potentials in trapped ions

“…The creation of an electrostatic potential via a detuned laser acting on ion 2 yields the interesting case of interaction terms η Ω2 xσ x 2 /∆ + Ω 2 σ z 2 . For a large detuning Ω 2 ≫ η Ω2 this interaction becomes effectively ∝ x2 σ z 2 , allowing the simulation of quadratic potentials [14].…”