Perturbation Theory for the Bound Polaron

Sak and Wang, Arora, and Matsuura have recently calculated the ground-state energy of a bound polaron in second-order perturbation theory by making use of an identity that for a Hermitian operator $O〈f\left|O\right|g〉=〈g\left|O^{*}\right|f〉$. It is the purpose of the present communication to show that the identity is incorrectly used by these authors since one of the integrals involved in the identity is divergent and in this situation the identity does not hold. Fortunately, this does not significantly affect the specific conclusions of their work, but amends the analysis in arriving at their result.