Sergio Albeverio, Yuri Kondratiev, Yuri Kozitsky.
Nonlinear S-transform
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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.