Self-similarity in quantum dynamics

Summary

We have developed a renormalization transformation, based on the existence of higher-order nonlinear resonances in the double-resonance model, that gives good predictions for the extension of the wave function in that system due to nonlinear resonance overlap. The double-resonance model describes the qualitative behavior, in local regions of the Hilbert space, of many quantum systems with two degrees of freedom whose dynamics is described by a nonlinear Hamiltonian but a linear Schrödinger equation.

INTRODUCTION

The phase space of nonlinear nonintegrable classical conservative systems exhibits extremely complex structure consisting of regular Kolmogorov-Arnold-Moser (KAM) tori intermixed with chaos. In some regions of the phase space the structure is self-similar to all length scales and exhibits scaling behavior in space and time. KAM tori are the remnants of global conserved quantities. For many systems, when some parameter which characterizes the size of the nonlinearity is small, KAM tori dominate the phase space. However, as the nonlinearity parameter is increased in size, nonlinear resonances in the system grow and overlap and destroy KAM tori lying between them. The existence of KAM tori can have a profound effect on the dynamics of a conservative system with two degrees of freedom because some KAM tori can divide the phase space into disjoint parts. When such a KAM torus is destroyed by nonlinear resonance overlap, the dynamics of the classical system may change dramatically.

The mechanism by which KAM surfaces are destroyed by nonlinear resonances has been studied extensively by Greene, Shenker and Kadanoff, MacKay, and others. KAM tori have irrational winding numbers. Each irrational winding number can be represented uniquely by a continued fraction.