Lee-Yang measures and wave functions

Abstract: We establish necessary and sufficient conditions for a Borel measure to be a Lee-Yang one which means that its Fourier transform possesses only real zeros. Equivalently, we answer a question of P\'olya who asked for a characterisation of those positive positive, even and sufficiently fast decaying kernels whose Fourier transforms have only real zeros. The characterisation is given in terms of Wronskians of polynomials that are orthogonal with respect to the measure. The results show that Fourier transforms of a rather general class of measures can be approximated by symmetrized Slater determinants composed by orthogonal polynomials, that is, by some wave functions which are symmetric like the Boson ones. Brief comments on possible interpretation and applications of the main results in quantum and statistical mechanics, to Toda lattices and the general solution of the heat equation, are given. We discuss briefly the possibility of represent the Riemann $\xi$ function as a partition function of a statistical mechanics system.