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Stable directions for excited states of nonlinear Schrödinger equations. (English) Zbl 1021.35113

Summary: We consider nonlinear Schrödinger equations in \(\mathbb{R}^3\). Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-selfadjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although selfadjoint perturbation turns embedded eigenvalues into resonances, this class of non-selfadjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden rule.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81U05 \(2\)-body potential quantum scattering theory
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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