非线性薛定谔方程沿曲线携带动量的解。部分II：存在性结果的证明
Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result
摘要：
本文证明了（椭圆型）非线性Schroedinger方程$- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$在流形上或欧几里得空间中一类特殊解的存在性。这里V表示电势，p是大于1的指数，而$ε$是对应于普朗克常数的小参数。当$ε$趋于零（即在半经典极限）时，我们证明了复值解的存在性，这些解集中在闭曲线上，并且其相位是高度振荡的.物理上，这些解沿着极限曲线携带量子力学动量。在本文的第一部分中，我们确定了极限集并构造了近似解，而在这里，我们给出了我们的主要存在结果的完整证明。
英文摘要：
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $ε$ a small parameter corresponding to the Planck constant. As $ε$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this work we identified the limit set and constructed approximate solutions, while here we give the complete proof of our main existence result.
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