99-253 Sergio Albeverio, Yuri Kondratiev, Yuri Kozitsky

Nonlinear $S$-transform and Critical Point Convergence for a Quantum Hierarchical Model (67K, Latex) Jul 1, 99

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Abstract. A sequence of measures $\{\nu_n \}$ on a separable Hilbert space ${\cal H}$ generated by a nonlinear map is considered. For a special choice of ${\cal H}$ and $\nu_0$, such a sequence describes the Euclidean Gibbs states of a chain of interacting quantum anharmonic oscillators. Each ${\nu_n }$ has a Laplace transform $F_n$, which is an entire function on ${\cal H}$. The sequence $\{F_n \}$ can be generated by a nonlinear generalization of the $S$-transform known in Gaussian Analysis, defined as a holomorphic map on certain spaces of entire functions. A family of fixed points for this map is found and analyzed. In the case where $\{F_n \}$ describes the mentioned oscillators, it is proven that this sequence converges to both stable and unstable fixed point. The convergence to the unstable fixed point corresponds to the appearance of the strong dependence between the oscillators peculiar to the critical point.

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