Spectral Characterization, Local Rigidity and Morse Index of Bubble Solutions to the CR Fractional Yamabe Problem

We study bubble solutions to the critical CR fractional Yamabe equation on the Heisenberg group for the full range \(0<γ<\frac{Q}{2}\), including the admissible integer-order cases. The CR fractional operator is genuinely nonlocal and is described by Gamma-function spectral multipliers. Its structure differs from that of the Euclidean fractional Laplacian, so the linearized analysis cannot be obtained by a direct extension of integer-order arguments. We develop a direct spectral approach based on the Cayley intertwining relation and the complex spherical-harmonic decomposition, which characterizes the kernel of the linearized operator in terms of infinitesimal translation and dilation modes without reducing the equation to an integral equation or requiring a separate decay analysis. As a byproduct, we obtain the exact normalization constant of the standard bubble throughout the admissible range. We then prove three main results: $(i)$ the non-degeneracy of the bubble manifold; $(ii)$ the local rigidity of the bubble manifold, via a uniform invertibility estimate on the orthogonal complement of the tangent space, with an explicit optimal constant; $(iii)$ the Morse index of every positive bubble is exactly one. The results provide a unified spectral and variational framework for the CR Yamabe problem on the Heisenberg group, covering both integer and noninteger orders.