Correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundaries

We study field correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundaries. We derive the Fredholm minor determinant formulas for the correlation functions in a finite box. In the thermodynamic limit, we express the correlation functions in terms of solutions of non-linear differential equations which was introduced by M. Jimbo, T. Miwa, Y. M\^ori and M. Sato as the generalization of the fifth Painlev\'e equations