Nonlinear S-transform

A nonlinear generalization of the S-transform known in Gaussian analysis is introduced. It is deened as a nonlinear operator on topolog-ical spaces of entire functions of innnitely many variables with growth of at most second order and nite type restricted to a certain value. This nonlinear S-transform (NLST) has xed points a certain 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 speciied here to describe the temperature Gibbs states of hierarchically interacting quantum anharmonic oscil-lators. Two families of models of such oscillators { M + and M ? , 1 are considered. In the case of M ? , it is proven that for all values of the model parameters, the convergence occurs only to the stable xed point which corresponds to the absence of any critical point anomalies for such models at all temperatures. In the case of M + , however, it is proven that varying the parameters of the model within the frames of the family one may obtain the convergence to both types of xed points { stable and unstable. The convergence to the unstable xed point corresponds to the appearance of a strong dependence between the oscillators which is peculiar to the critical point of the model.