HIROSHIMA, F., LORINCZI, J. and TAKAESU, T., 2012. A probabilistic representation of the ground state expectation of fractional powers of the boson number operator. Journal of Mathematical Analysis and Applications, 395 (2), pp.437-447
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.