Strong Instability of Standing Waves for Nonlinear Schrödinger Equations with Harmonic Potential

Abstract

We study strong instability of standing waves e^iωtφ_ω(x) for nonlinear Schrödinger equations with L²-supercritical nonlinearity and a harmonic potential, where φ_ω is a ground state of the corresponding stationary problem. We prove that e^iωtφ_ω(x) is strongly unstable if ∂_λ²E(φ_ω^λ)|_λ=1 ≤ 0, where E is the energy and v^λ(x) = λ^N/2v(λx) is the L²-invariant scaling.