Interacting Fock Spaces Related to the Anderson Model

A new type of interacting Boltzmannian Fock space, emerging from the stochastic limit of the Anderson model, is investigated. We describe the structure of the space and the form, assumed in this case, by the principles of factorization and of total connection. Using these principles, the vacuum expectation of any product of creation and annihilation operators can be calculated. By means of these results, for any test function, a system of diffrence equations satisfied by the moments of the field operator and an integral equation satisfied by their generating function is deduced. In many interesting cases this equation is solved and the vacuum distribution function of the field operator (even its density) is explicitly determined. This evidentiates a new phenomenon which cannot take place in the usual Fock spaces (and did not appear in the simplest examples of interacting Fock spaces): by taking different test functions, the vacuum distribution of the field operator does not change only parametrically, but radically. In particular we find the semi-circle, the reciprocal-semi-circle (or Arcsine), the double-beta,…, and many other distributions.