发表机构
Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对三维三次-五次Klein-Gordon和Schrödinger方程基态，研究其线性化算子的离散谱，证明角动量扇区数趋于无穷并给出尖锐渐近，且证明内部模式存在于特定区间，为非线性稳定性分析奠定基础。
AI 中文摘要
受三维空间中Klein-Gordon和Schrödinger方程基态在存在内部模式情况下的渐近稳定性问题启发，我们研究了具有三次-五次非线性的相应模型。这些方程出现在多个物理背景中；特别是，三次-五次Klein-Gordon方程自然地出现在量子场论中自旋-$0$粒子的研究中。更精确地说，设$Q_\omega$为三维三次-五次椭圆方程\\[ -\Delta Q_\omega+\omega Q_\omega-Q_\omega^3+Q_\omega^5=0 \\]的正径向基态。我们研究两个相关但逻辑上不同的谱问题，当$\omega$趋近于基态分支的端点$3/16$时。首先，我们确定标量Hessian算子\\[ L_+=-\Delta+\omega-3Q_\omega^2+5Q_\omega^4 \\]的完整离散谱。该算子也是质量参数为$\omega$的实标量三次-五次Klein-Gordon方程在静态态$Q_\omega$附近的精确线性化空间算子。我们证明其离散角动量扇区的数量趋于无穷，并获得其位置和重数的尖锐渐近。其次，对于三次-五次非线性Schrödinger方程，我们分析完整的Hamiltonian矩阵线性化，并证明在长度与$(3/16-\omega)^{-1}$相当的角动量扇区区间内恰好存在内部模式。这些结果为三维三次-五次模型未来的非线性稳定性和辐射阻尼分析提供了严格的线性基础。
英文摘要
Motivated by the problem of asymptotic stability for ground states of the Klein--Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic--quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic--quintic Klein--Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let $Q_ω$ be the positive radial ground state of the three-dimensional cubic--quintic elliptic equation \[ -ΔQ_ω+ωQ_ω-Q_ω^3+Q_ω^5=0. \] We study two related, but logically distinct, spectral problems as $ω$ approaches the endpoint $3/16$ of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian \[ L_+=-Δ+ω-3Q_ω^2+5Q_ω^4. \] This operator is also the exact linearized spatial operator around the static state $Q_ω$ for the real scalar cubic--quintic Klein--Gordon equation with mass parameter $ω$. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic--quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to $(3/16-ω)^{-1}$. %The rotating complex Klein--Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic--quintic models.
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