分数阶薛定谔方程多峰集中解的Morse指数、Leray-Schauder度与局部唯一性
Morse index, Leray-Schauder degree and local uniqueness for multi-peak concentrating solutions of a fractional Schrödinger equation
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中文总结 AI 辅助
该论文针对半经典分数阶薛定谔方程的多峰集中解，确定线性化算子低谱与Morse指数，结合调制参数化与度计算证明小ε下对应集中类含唯一正解。
中文摘要 AI 辅助
我们研究了半经典分数阶薛定谔方程ε^(2s)(-Δ)^s u + V(x)u = u^p在ℝ^N中的正k峰解，这类解集中在V的不同非退化临界点ξ₁⁰,…,ξ_k⁰处。对于每个满足自然能量量子化的此类解族，我们确定了线性化算子的完整低谱：前k个特征值保持一致为负，接下来的kN个特征值阶为ε²，由Hessian矩阵D²V(ξ_j⁰)控制，其余谱与零一致分离。因此，Morse指数等于k加上这些Hessian矩阵的负特征值总数，且每个此类解是非退化的。结合唯一的调制参数化与Leray-Schauder度计算，我们进一步证明，对于所有足够小的ε，给定的集中类恰好包含一个正解。该结果适用于整个能量量子化类，而非仅特定解。
英文摘要
We study positive $k$-peak solutions of the semiclassical fractional Schrödinger equation $\varepsilon^{2s}(-Δ)^s u+V(x)u=u^p$ in $\mathbb{R}^N$, concentrating at different nondegenerate critical points $ξ_1^0,\ldots,ξ_k^0$ of $V$. For every such family satisfying the natural energy quantization, we determine the complete low spectrum of the linearized operator. The first $k$ eigenvalues remain uniformly negative, the next $kN$ eigenvalues are of order $\varepsilon^2$ and are governed by the Hessians $D^2V(ξ_j^0)$, while the remaining spectrum is uniformly separated from zero. Consequently, the Morse index equals $k$ plus the total number of negative eigenvalues of these Hessians, and every such solution is nondegenerate. Combining a unique modulation parametrization with a Leray--Schauder degree computation, we further prove that, for all sufficiently small $\varepsilon$, the prescribed concentrating class contains exactly one positive solution. The result applies to the whole energy-quantized class, not only to a particular solution.
