[Paper Review] Asymptotic Dynamics of Nonlinear Schrödinger Equations with Many Bound States

Tai‐Peng Tsai|ArXiv.org|Apr 30, 2002

Advanced Mathematical Physics Problems12 references4 citations

TL;DR

This paper establishes the asymptotic convergence of small, localized solutions to the nonlinear Schrödinger equation in $\mathbb{R}^3$ with a potential possessing three or more bound states. Under resonance conditions and initial data with a dominant ground state component exceeding $n^{3-\epsilon}$, the solution converges locally to a nonlinear ground state as $t \to \infty$, with decay rates governed by dispersive estimates and spectral projections, extending prior results from one- or two-bound-state systems.

ABSTRACT

We consider a nonlinear Schrödinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order $n$ in $H^1$, and that its ground state component is larger than $n^{3-ε}$ with $ε>0$ small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity.

Motivation & Objective

• To understand the long-time behavior of solutions to the nonlinear Schrödinger equation when the linear Hamiltonian has three or more bound states.
• To extend previous results on asymptotic stability of nonlinear ground states from systems with one or two bound states to those with $N+1$ bound states, $N \geq 2$.
• To identify conditions under which solutions with small, localized initial data converge locally to a nonlinear ground state as $t \to \infty$, despite the presence of excited states.
• To analyze the role of resonance between bound states and the continuous spectrum in facilitating the decay of excited state components.

Proposed method

• The analysis relies on a decomposition of the solution into ground state, excited state, and remainder components using spectral projections associated with the linear Hamiltonian $H_0 = -\Delta + V$.
• A key technique involves the use of the linearized operator $\mathcal{L}_E$ around a nonlinear ground state $Q_E$, enabling the study of perturbations via the evolution $e^{t\mathcal{L}}$.
• Dispersive estimates in $L^5$ and $L^2_{\mathrm{loc}}$ norms are applied to control the remainder term $\xi(t)$, with decay rates $t^{-9/10}$ and $t^{-3/2}$ respectively.
• The proof uses a time-splitting argument: first, the solution is evolved to a time $t_2$ where the ground state component dominates, then the dynamics are analyzed from $t_2$ onward using a refined decomposition.
• Resonance conditions on the eigenvalues $e_0 < e_1 < \cdots < e_N$ are assumed to ensure the persistence of decay mechanisms from excited states into the continuous spectrum.
• The analysis incorporates $W^{k,p}$ estimates for the wave operator and assumes $V$ satisfies conditions ensuring the absence of embedded eigenvalues or resonances at the bottom of the continuous spectrum.

Experimental results

Research questions

• RQ1 Under what conditions does a solution to the nonlinear Schrödinger equation with multiple bound states converge locally to a nonlinear ground state as $t \to \infty$?
• RQ2 How does the presence of multiple excited states affect the asymptotic decay rate of the solution's local $L^2$-norm difference from the ground state?
• RQ3 Can the dynamics be controlled when the initial data are not near a nonlinear ground state, but still small and localized?
• RQ4 What role do spectral resonance conditions among the bound state energies play in enabling the decay of excited state components?
• RQ5 How does the size of the initial ground state component ($> n^{3-\epsilon}$) influence the convergence to a nonlinear ground state?

Key findings

• Solutions with initial data in $H^1$ of size $n$ and a ground state component larger than $n^{3-\epsilon}$ for small $\epsilon > 0$ converge locally to a nonlinear ground state as $t \to \infty$.
• The local $L^2$-norm difference between the solution and the nonlinear ground state decays as $t^{-1/2}$, slower than the $t^{-3/2}$ rate in the one-bound-state case due to the persistence of excited states.
• The remainder term $\xi(t)$, representing the deviation from the ground state, satisfies $\|\xi(t)\|_{L^2_{\mathrm{loc}}} \lesssim n^3 t_2 (1+t-t_2)^{-1/2}$, indicating dispersive decay over time.
• The asymptotic profile is a nonlinear ground state $Q_{E_\infty}$, with $E_\infty$ close to the initial energy, and the convergence is robust under small perturbations of the initial data.
• The analysis confirms that the solution does not remain trapped in excited state manifolds; instead, the dynamics are driven toward the ground state due to resonance-induced decay into the continuous spectrum.
• The decay of the excited state components is governed by dispersive estimates with $L^5$ and $L^2_{\mathrm{loc}}$ norms, yielding bounds of order $t^{-9/10}$ and $t^{-3/2}$ respectively, which are critical for controlling the perturbation.