能量临界量子谐振子
The Energy-Critical Quantum Harmonic Oscillator
摘要：
本文考虑具有谐振子势$V(x) = \tfrac{1}{2} |x|^2$的$d \ge 3$维能量临界非线性薛定谔方程.当非线性是散焦，我们证明了全球适定性的所有初始数据在能量空间$Σ$，包括所有功能$u_0$，使$\nabla u_0$和$x u_0$属于$L^2$。这一结果推广了Killip-Visan-Zhang \cite{kvz_quadratic_potentials}的一个定理，该定理处理径向情形。对于聚焦问题，我们得到的所有数据满足模拟通常的大小限制的基态$W$的全球适定性。证明使用能量归纳范式的浓度紧凑性变体。特别是，我们开发了一个线性剖面分解适应传播$\exp[ it(\tfrac{1}{2}Δ- \tfrac{1}{2}|x|^2)]$有界序列$Σ$。
英文摘要：
We consider the energy critical nonlinear Schrödinger equation in dimensions $d \ge 3$ with a harmonic oscillator potential $V(x) = \tfrac{1}{2} |x|^2$. When the nonlinearity is defocusing, we prove global wellposedness for all initial data in the energy space $Σ$, consisting of all functions $u_0$ such that both $\nabla u_0$ and $x u_0$ belong to $L^2$. This result extends a theorem of Killip-Visan-Zhang \cite{kvz_quadratic_potentials}, which treats the radial case. For the focusing problem, we obtain global wellposedness for all data satisfying an analogue of the usual size restriction in terms of the ground state $W$. The proof uses the concentration compactness variant of the induction on energy paradigm. In particular, we develop a linear profile decomposition adapted to the propagator $\exp[ it(\tfrac{1}{2}Δ- \tfrac{1}{2}|x|^2)]$ for bounded sequences in $Σ$.
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