The aim of this paper is to investigate the qualitative properties of the multi-peak solution \( u_\varepsilon \) to the following Schrödinger-Poisson problem: \(\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^{2}\Delta u + V(y)u + \Phi (y)u = |u|^{p-1}u, & y \in \mathbb {R}^3, \\ -\Delta \Phi (y) = u^2, & y \in \mathbb {R}^3, \end{array}\right. } \end{aligned}\) where \( \varepsilon > 0 \) is a small parameter, \( V(y) \) is a potential function, and \( 1< p < 5 \) . Under the assumption that \( \{P_i\}_{i=1}^{m} \) are the non-degenerate critical points of \( V(y) \) , we derive an explicit formula for the Morse index of \( u_\varepsilon \) , closely related to the negative eigenvalues of the Hessian matrix \( D^2 V(P_i) \) . As a consequence, we also establish the non-degeneracy of \( u_\varepsilon \) . Unlike the classical Schrödinger equation, the presence of the non-local term \( \Phi (y) \) introduces significant challenges in analyzing the asymptotic behavior of the eigenpairs associated with the linearized operator at \( u_\varepsilon \) , which require precise and advanced analytical techniques.
