Generalized Kac Problem for Bose Gas in Polygonal Region

摘要

The paper considers interacting Bose gas in a polygonal domain and studies the asymptotics of the log-partition function in its Feynman-Kac representation as the domain is delated to infinity. It is proved that for repulsive interaction with power decay at infinity the asymptotics of the log-partition function determines the area of the domain, the length of its boundary and the constant term defined by the angles of the polygon. This is a natural generalization of the Kac's famous problem on computing the asymptotics of the partition function Tre~(βΔ), where Δ is the Dirichlet Laplacian for a polygonal domain.