Polarons Bound in a Coulomb Potential. I. Ground State

We introduce a trial function for the ground state of a polaron bound in a Coulomb field which yields, in the experimentally important ranges of Coulomb binding and polaron coupling constant $\alpha$ lower energies than have previously been reported. In distinction from earlier calculations, our ansatz gives, additionally, the correct polaron mass renormalization in the weak-binding-weak-coupling limit. For the very weakly bound polaron, our trial function is not continuous in $\alpha$; a discontinuity occurs at $6.25<\mathrm{}\alpha <6.5\mathrm{}$. We speculate that this discontinuity is associated with a long conjectured breakdown of perturbation theory for the free-polaron ground state near $\alpha =6\mathrm{}$. Methods for evaluating the perturbed bound-polaron ground-state energy for weak coupling are discussed.