Strongly interacting bumps for the Schrodinger-Newton equations

We study concentrated bound states of the Schrodinger-Newton equations h(2) Delta psi -E(x)psi+U psi=0, psi > 0, x is an element of R-3; Delta U+1/2 vertical bar psi vertical bar(2)=0, x is an element of R-3; psi(x)-> 0, U(x)-> 0 as vertical bar x vertical bar ->infinity. Moroz et al. [An analytical approach to the Schrodinger-Newton equations, Nonlinearity 12, 201 (1999)] proved the existence and uniqueness of ground states of Delta psi-psi+U psi=0, psi > 0, x is an element of R-3; Delta U+1/2 vertical bar psi vertical bar(2)= 0, x is an element of R-3; psi(x)-> 0, U(x)-> 0 as vertical bar x vertical bar ->infinity. We first prove that the linearized operator around the unique ground state radial solution (psi(0), U-0) with psi(0)(r)=(Ae(-r)/r)(1+ o(1)) as r=vertical bar x vertical bar ->infinity, U-0(r) =(B/r)(1+ o(1)) as r=vertical bar x vertical bar ->infinity for some A, B > 0 has a kernel whose dimension is exactly 3 (corresponding to the translational modes). Using this result we further show that if for some positive integer K the points P-i is an element of R-3, i= 1,2 . . . , K, with P-i not equal P-j for i not equal j are all local minimum or local maximum or nondegenerate critical points of E(P), then for h small enough there exist solutions of the Schrodinger Newton equations with K bumps which concentrate at Pi. We also prove that given a local maximum point P-0 of E(P) 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)). (C) 2009 American Institute of Physics. [DOI: 10.1063/1.3060169]