INTERTWINING SEMICLASSICAL SOLUTIONS TO A SCHRODINGER-NEWTON SYSTEM

We study the problem {(-epsilon i del + A(x))(2) u + V(x)u = epsilon(-2) (1/vertical bar x vertical bar * vertical bar u vertical bar(2)) u, u is an element of L-2(R-3, C), epsilon del u + iAu is an element of L-2 (R-3, C-3), where A: R-3 -> R-3 is an exterior magnetic potential, V: R-3 -> R is an exterior electric potential, and epsilon is a small positive number. If A = 0 and epsilon = h 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 effect. We assume that A and V are compatible with the action of a group G of linear isometrics of R-3. Then, for any given homomorphism T: G -> S-1 into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential V on the number of semiclassical solutions u : R-3 -> C which satisfy u(gx) = T(g) u(x) for all g is an element of G, x is an element of R-3. We also study the concentration behavior of these solutions as epsilon -> 0.