Stable Directions for Excited States of Nonlinear Schrödinger Equations

2001/10/30 by Tai‐Peng Tsai, Tai-Peng Tsai, Horng-Tzer Yau +3 · 2 citations
Mathematics · Physics and Astronomy · #35Q40 #35Q55 #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:35Q40 #msc:35Q55

paper · pdf · doi:10.48550/arxiv.math-ph/0110037

v2: sec 3 changed

openalex publication_date 2001/10/30 · arxiv created 2002/01/02 · arxiv updated 2016/09/07 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We consider nonlinear Schrödinger equations in \R³. 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.

Citations

• Spectral properties of Schrödinger operators and time-decay of the wave functions
• Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations
• Asymptotic dynamics of nonlinear Schrödinger equations: Resonance‐dominated and dispersion‐dominated solutions

Cited by

• On instability of excited states of the nonlinear Schrödinger equation
• Spectral Theory and Nonlinear PDE: a Survey

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• Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations
• Relaxation of Excited States in Nonlinear Schrödinger Equations
• Small solutions of nonlinear Schrödinger equations near first excited states
• On instability of excited states of the nonlinear Schrödinger equation
• Asymptotic Dynamics of Nonlinear Schrodinger Equations: Resonance Dominated and Radiation Dominated Solutions