Generalization of the hypervirial and Feynman-Hellman theorems

TL;DR

This paper extends the hypervirial and Feynman-Hellman theorems to singular potentials and general second-order differential equations, highlighting their physical implications and differences.

Contribution

It generalizes the theorems to singular potentials and broad classes of equations, providing new insights into their physical applications.

Findings

Generalized theorems to singular potentials

Analyzed physical consequences of the generalization

Discussed differences with Feynman-Hellmann theorems

Abstract

Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is carried on for most general second order differential equation, which involves all physically interesting cases, such as Schr\"odinger and two-body Klein-Gordon equations with singular potentials. Some physical consequences are discussed. The connection with Feynman-Hellmann like theorems are also considered and some relevant differences are underlined.