The Complex-Germ Method for Statistical Mechanics of Model Systems

We consider a special (“semiclassical”) class of Hamiltonian operators in the Fock space. This class involves the superconductivity BCS model, as well as the lattice magnetic Hamiltonians. It is shown that the commutator of two observables from this class is proportional to a small parameter. For each observable of this type, one can introduce a function on the phase space which corresponds to the observable. The Poisson bracket between two functions is defined. Two approaches are developed for investigating the Schrödinger equation with the Hamiltonian of the class under consideration. One of them is based on the Ehrenfest theorem, and another is based on the substitution of the hypothetical asymptotic solution to the equation. We construct the “semiclassical” states obeying the following property: in the “semiclassical” limit, the average values of “semiclassical” observables in such states coincide with the values of the corresponding functions at a certain point of the classical phase space. The asymptotics constructed can be interpreted in terms of the complex-germ theory.