STABLE DIRECTIONS FOR EXCITED STATES OF NONLINEAR SCHRÖDINGER EQUATIONS

Abstract

ABSTRACT We consider nonlinear Schrödinger equations in . 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-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint 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. Keywords: Stable directionExcited stateSchro¨dinger equationEmbedded eigenvalueResonanceFermi golden rule ACKNOWLEDGMENTS The authors would like to thank L. Erdös and S. Cuccagna for their very helpful comments and discussions. Part of this work was done when both authors visited the Academia Sinica and the Center for Theoretical Sciences in Taiwan. Their hospitalities are gratefully acknowledged. Tsai was partially supported by NSF grant DMS-9729992. Yau was partially supported by NSF grant DMS-0072098.

Locations

• Communications in Partial Differential Equations - View
• arXiv (Cornell University) - View - PDF
• Digital Access to Scholarship at Harvard (DASH) (Harvard University) - View - PDF