Intertwining semiclassical solutions to a schrodinger-newton system

We study the problem( (ir + A(x))2 u + V (x)u = 2 - 1 jxj - juj2 - u; u 2 L2(R3;C); "ru + iAu 2 L2(R3;C3); where A: R3 ! R3 is an exterior magnetic potential, V : R3 ! R is an exterior electric potential, and " is a small positive number. If A = 0 and " = ~ is Planck's constant this problem is equivalent to the Schrodinger-Newton equations proposed by Penrose in [23] to describe his view that quantum state reduction occurs due to some gravitational e-ect. We assume that A and V are compatible with the action of a group G of linear isometries of R3. Then, for any given homomorphism - : G ! S1 into the unit complex numbers, we show that there is a combined e-ect of the symmetries and the potential V on the number of semiclassical solutions u : R3 ! C which satisfy u(gx) = -(g)u(x) for all g 2 G, x 2 R3. We also study the concentration behavior of these solutions as " ! 0:.