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Selection of the ground state for nonlinear Schrödinger equations. (English) Zbl 1111.81313

Summary: We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree-Fock type.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35Q55 NLS equations (nonlinear Schrödinger equations)
81U05 \(2\)-body potential quantum scattering theory
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