Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics

Abstract

We show, by making use of the functional integral technique, that, for a large class of useful quantum statistical systems, the partition function is, with respect to the coupling constant, the Laplace transform of a positive measure. As a consequence, we derive an infinite set of monotonicly converging upper and lower bounds to it. In particular, the lowest approximation appears to be identical to the Gibbs–Bogolioubov variational bound, while the next approximations, for which we give explicit formulas for the first few ones, lead to improve the previous bound. The monotonic character of the variational successive approximations allows a new approach towards the thermodynamical limit.

Perturbation of Wigner matrices and a conjecture

“…If this is the case, then µ depends both on the spectra of the two matrices and on the relative position of their eigenvectors. The function (1), and especially its derivatives at t = 0, define important quantities in quantum statistical mechanics. Proving positive definiteness would lead to interesting relations among them.…”

Positive eigenvalues of generalized words in two Hermitian positive definite matrices

“…We call such a g-word good and all other g-words bad. Interest in this problem stems from a question in quantum physics [1,7], as discussed in [5] for the case of (ordinary) words (p i , q i positive integers) and real symmetric positive definite matrices. For 2-by-2 matrices, the situation is better understood.…”

On the function TreH+itK

“…It is a widely known conjecture [1,4,8] ( 1.2) is positive semidefinite or, equivalently, that there exists a measure µ on R such that Tr e H+itK = e itx dµ(x) (t ∈ R).…”