SCHR¨ODINGER AND DIRAC PARTICLES IN QUASI-ONE-DIMENSIONAL SYSTEMS WITH A COULOMB INTERACTIONстатья
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Аннотация:We consider specific features and principal distinctions in the behavior of the energy spectra of Schr¨odinger
and Dirac particles in the regularized “Coulomb” potential Vδ(z) = −q/(|z|+δ) as functions of the cutoff
parameter δ in (1+1) dimensions. We show that the discrete spectrum becomes a quasiperiodic function of
δ for δ 1 in such a one-dimensional “hydrogen atom” in the relativistic case. This effect is nonanalytically
dependent on the coupling constant and has no nonrelativistic analogue in this case. This property of the
Dirac spectral problem explicitly demonstrates the presence of a physically informative energy spectrum
for an arbitrarily small δ > 0, but also the absence of a regular limit transition δ → 0 for all nonzero q.
We also show that the three-dimensional Coulomb problem has a similar property of quasiperiodicity with
respect to the cutoff parameter for q = Zα > 1, i.e., in the case where the domain of the Dirac Hamiltonian
with the nonregularized potential must be especially refined by specifying boundary conditions as r → 0
or by using other methods.
Keywords: relativistic effect, Dirac equation, regularized Coulomb potential, one-dimensional hydrogen
atom