Strongly interacting bumps for the schrodinger-newton equations

Abstract

We study concentrated bound states of the Schrodinger-Newton equations Moroz, Penrose and Tod proved the existence and uniqueness of ground states.

We first prove that the linearized operator around the unique ground state radial solution has a kernel whose dimension is exactly 3 (correspondingto the translational modes).

Using this result we further show: If for some positive integer K the points P_i in R^3, i=1,2,...,K$ with P_i\not=P_j for i\not=j are all local minimum or local maximum or nondegenerate critical points of the reduced energy function then forn h small enough there exist solutions of the Schrodinger-Newton equations with K bumps which concentrate at P_i.

We also prove that given a local maximum point P_0 of the reduced energy there exists a solution with K bumps which all concentrate at P_0 and whose distances to P_0 are at least O(h^(1/3))