Abstract

A single particle is bound by an attractive central potential and obeys the Dirac
equation in d spatial dimensions. The Coulomb potential is one of the few examples
for which exact analytical solutions are available. A geometrical approach called \the
potential envelope method" is used to study the discrete spectra generated by potentials
V (r) that are smooth transformations V (r) = g(-1/r) of the soluble Coulomb
potential. When g has de�nite convexity, the method leads to energy bounds. This
is possible because of the recent comparison theorems for the Dirac equation. The
results are applied to study soft-core Coulomb potentials used as models for con-
�ned atoms. The spectral estimates are compared with accurate eigenvalues found
by numerical methods.
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