Nondegeneracy of solutions to nonlinear elliptic PDEs plays an important role in the construction of new solutions. For Schrödinger type systems, proving nondegeneracy is usually quite challenging in general situations. In this paper, we continue our previous work [23], and focus on nondegeneracy results of \(n(n\ge 3)\) -coupled Schrödinger systems. By computing the eigenvalues of some highly complex nth-order real symmetric matrices, using orthogonal transformations and decomposing the Laplace operator in \(\mathbb {R}^N\) into radial and spherical parts, we successfully establish the nondegeneracy of the positive solutions to n-coupled Schrödinger systems for \(n\ge 3\) , which generalizes the results of [E. N. Dancer and J. Wei, Trans. Amer. Math. Soc. 2009], where nondegeneracy of positive solutions for the coupled Schrödinger system was established specifically for the case n=2. As an application, we construct a family of new non-radial positive solutions and show uniqueness of single peak solutions for the following multi-species nonlinear Schrödinger system under different trapping potentials \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_{i}+V(x)u_{i}=a_{i}u^{3}_{i}+\sum ^{n}_{i\ne j,j=1}\beta u^{2}_{j}u_{i},\ \text {in} \ \mathbb {R}^{N},\\ u_{i}>0\ \text {in} \ \ \mathbb {R}^{N}, \ u_{i}\rightarrow 0 \ \text {as} \ |x|\rightarrow +\infty ,i=1,2,\cdots ,n. \end{array}\right. } \end{aligned}\) First, by using building blocks different from the ones adopted by [ S. Peng and Z.-Q. Wang, Arch. Ration. Mech. Anal. 2013], we successfully construct new synchronized vector solutions for \(n(n\ge 3)\) -coupled Schrödinger system with more complex concentration structure. Then, we prove that the single peak solutions, if they concentrate on the same point, are unique. As far as we know, this is the first systematic study uniqueness result for single peak solutions of \(n(n\ge 3)\) -coupled nonlinear Schrödinger systems under radially symmetric but non-monotonic potentials.
