Abstract
The quantal system of Bose particles described by the non-linear Schrödinger equation i ∂φ/ ∂t = - 1 2 ∂ 2φ/∂x 2 + cφ∗φ 2, with c= cxf∞ and via the ground state with finite particle density, is the 1- dimensional gas of impenetrable bosons studied by M. Girardeau, T.D. Schultz, A. Lenard, H.G. Vaidya and C.A. Tracy. We show that the 2-point (resp. 2 n-point) function, or the 1-particle (resp. n-particle) reduced density matrix, of this system satisfies a non-linear differential equation (resp. a system of non-linear partial differential equations) of Painlevé type. Derivation of these equations is based on the link between field operators in a Clifford group and monodromy preserving deformation theory, which was previously established and applied to the 2-dimensional Ising model and other problems. Several related topics are also discussed.
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