 99253 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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