In the first part we consider the Laplace operator with Neumann boundary conditions on a configuration space with Poisson measure over a bounded domain. The spectrum of this operator is considered and the structure of its vacuum space is studied. The corresponding spectral gap inequality is proved. The differences between Poincare and spectral gap inequalities are shown, and absence of Poincare inequality is presented. In the second part we study a second order differential operator with grown coefficients on a whole configuration space. The main properties of this operator are considered and <$E roman {Poncar e back 30 up 30 symbol В}> inequality is proved.

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