Distribution functions in virial and hypervirial theory

The forms of the virial and hypervirial theorems are derived when a distribution function, that is, the Laplace transform of a distribution is taken to define a quantum-mechanical system. The validity of the resulting equations imposes necessary (and sufficient) conditions on the distribution and hence the distribution function. Distribution theory is used to establish the existence of an exponential function satisfying the virial theorem for the Yukawa potential for a restricted range of the defining parameters. For hypervirial relations, the method would seem of little practical importance.