## 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
Full Text:

### References:

 [1] Agmon S., Ann. Scuola Norm. Sup. Pisa. Ser. IV 2 pp 151– (1975) [2] Buslaev V.S., St. Petersburg Math J. 4 pp 1111– (1993) [3] Buslaev V.S., preprint [4] DOI: 10.1002/cpa.1018 · Zbl 1031.35129 [5] Cuccagna S., Preprint [6] DOI: 10.1002/cpa.3160410602 · Zbl 0632.70015 [7] DOI: 10.1002/cpa.3160430302 · Zbl 0731.35010 [8] DOI: 10.1215/S0012-7094-79-04631-3 · Zbl 0448.35080 [9] DOI: 10.1002/cpa.3160440504 · Zbl 0743.35008 [10] DOI: 10.1007/BF01360915 · Zbl 0139.31203 [11] DOI: 10.1006/jdeq.1997.3345 · Zbl 0890.35016 [12] DOI: 10.1007/BF01609491 · Zbl 0381.35023 [13] Reed M., Methods of Modern Mathematical Physics II (1975) · Zbl 0308.47002 [14] Reed M., Methods of Modern Mathematical Physics IV (1975) · Zbl 0308.47002 [15] DOI: 10.1016/0167-2789(88)90107-8 · Zbl 0694.35202 [16] DOI: 10.1007/BF01212446 · Zbl 0603.35007 [17] DOI: 10.1090/S0002-9947-1985-0792821-7 [18] DOI: 10.1007/BF02096557 · Zbl 0721.35082 [19] DOI: 10.1007/s002220050303 · Zbl 0910.35107 [20] DOI: 10.1007/s000390050124 · Zbl 0917.35023 [21] DOI: 10.1002/cpa.3012 · Zbl 1031.35137 [22] DOI: 10.1155/S1073792802201063 · Zbl 1011.35120 [23] DOI: 10.1137/0516034 · Zbl 0583.35028 [24] DOI: 10.1002/cpa.3160390103 · Zbl 0594.35005 [25] DOI: 10.2969/jmsj/04730551 · Zbl 0837.35039 [26] Yajima K., J. Math. Sci. Univ. Tokyo 2 pp 311– (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.