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Relaxation of excited states in nonlinear Schrödinger equations. (English) Zbl 1011.35120

The paper considers the nonlinear Schrödinger equation with cubic nonlinearity, in the three-dimensional space, with an extra term accounting for an external potential (this equation is also frequently called a Gross-Pitaevskii equation, and it plays an important role in the theoretical description of the Bose-Einstein condensation). The authors consider a situation when the equation generates two nonlinear eigenstates with energies \(E_0 <E_1<0\) (the lower one is the system’s ground state, while the higher one is an excited state). Additionally, it is assumed that \(|E_0|> 2|E_1|\).
The objective of the work is to find out in what sense the excited state is unstable. It is demonstrated that solutions which stay close to it have zero measure in the space of the solutions, while all other solutions relax to the ground state via emission of radiation waves. The theorem is proved under certain conditions imposed on the initial state and the external potential.
In the course of the proof, three stages of the evolution of a solution which was originally close to the excited state are identified: (i) shedding off a radiation component, while the solution remains close to the excited state; (ii) the crucial stage, which is the relaxation from the excited state to the ground state; (iii) final asymptotic relaxation of the solution to the ground state. In order to handle the radiation-wave component of the solution, a definition of outgoing waves, valid for the nonlinear equation under consideration, is elaborated.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81R12 Groups and algebras in quantum theory and relations with integrable systems
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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