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Two zero-range-interacting atoms in a circular, transversely harmonicwaveguide are used as a test-bench for a quantitative description of thecrossover between integrability and chaos in a quantum system with no selectionrules. For such systems we show that the expectation value after relaxation ofa generic observable is given by a linear interpolation between its initial andthermal expectation values. The variable of this interpolation is universal; itgoverns this simple law to cover the whole spectrum of the chaotic behaviorfrom integrable regime through the well- developed quantum chaos. Thepredictions are confirmed for the waveguide system, where the mode occupationsand the trapping energy were used as the observables of interest; a variety ofthe initial states and a full range of the interaction strengths have beentested.

Memory of the Initial Conditions in an Incompletely-Chaotic Quantum System: Universal Predictions and an Application to Cold Atoms