A Probabilistic Representation of the Ground State Expectation of   Fractional Powers of the Boson Number Operator
  We give a formula in terms of a joint Gibbs measure on Brownian paths and the measure of a random-time Poisson process of the ground state expectations of fractional (in fact, any real) powers of the boson number operator in the Nelson model. We use this representation to obtain tight two-sided bounds. As applications, we discuss the polaron and translation invariant Nelson models. 
Fumio Hiroshima, Jozsef Lorinczi, Toshimitsu Takaesu
math-ph math.MP