Non-Degeneracy of Peak Solutions to the Schrodinger-Newton System

We are concerned with the following Schrodinger-Newton problem: -epsilon(2)Delta u + V(x)u = 1/8 pi epsilon(2) (integral(R3) u(2)(xi)/vertical bar x - xi vertical bar d xi)u, x is an element of R-3. For epsilon small enough, we prove the non-degeneracy of the positive solution to the above problem, that is, the corresponding linear operator L-epsilon(eta) = -epsilon(2)Delta eta(x) + V(x)eta(x) - 1/8 pi epsilon(2) (integral(R3)u(epsilon)(2)(xi)/vertical bar x - xi vertical bar d xi) eta(x) - 1/4 pi epsilon(2) (integral(R3)u(epsilon)(xi)eta(xi)/vertical bar x - xi vertical bar d xi)u(epsilon)(x) is non-degenerate, i.e., L-epsilon (eta(epsilon)) = 0 double right arrow eta(epsilon) = 0 for small epsilon > 0. The main tools are the local Pohozaev identities and the blow-up analysis. This may be the first non-degeneracy result on the peak solutions to the Schrodinger-Newton system.