97-273 Sergio Albeverio, Yuri Kondratiev, Yuri Kozitsky.

Nonlinear S-transform (74K, LaTeX) May 15, 97

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Abstract. A nonlinear generalization of the S-transform known in Gaussian analysis is introduced. It is defined as a nonlinear operator on topological spaces of entire functions of infinitely many variables with growth of at most second order and finite type. This nonlinear S-transform (NLST) has fixed points a family of which is found and analyzed. In particular, their stability is described in terms of spectral properties of the NLST Fr\'echet derivative. This is utilized to study the convergence of sequences of entire functions generated by the NLST, which is specified here to describe the temperature Gibbs states of models of hierarchically interacting quantum anharmonic oscillators. It is proven that the convergence to both types of fixed points -- stable and unstable -- holds provided the model parameters satisfy certain conditions. The convergence to the unstable fixed point corresponds to the appearance of a strong dependence between the oscillators which is peculiar to the critical point of the model.

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