Morse index, Leray-Schauder degree and local uniqueness for multi-peak concentrating solutions of a fractional Schr\"odinger equation
We study positive $k$-peak solutions of the semiclassical fractional Schr\"odinger equation $\varepsilon^{2s}(-\Delta)^s u+V(x)u=u^p$ in $\mathbb{R}^N$, concentrating at different nondegenerate critical points $\xi_1^0,\ldots,\xi_k^0$ 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 $D^2V(\xi_j^0)$, 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.