Intertwining semiclassical solutions to a Schrödinger-Newton system

We study the problem \[ \begin{cases} \left( -\varepsilon\mathrm{i}\nabla+A(x)\right) ^{2}u+V(x)u=\varepsilon ^{-2}\left( \frac{1}{|x|}\ast|u|^{2}\right) u,\\ u\in L^{2}(\mathbb{R}^{3},\mathbb{C}), \varepsilon\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), \end{cases} \] where A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3} is an exterior magnetic potential, V\colon\mathbb{R}^{3}\rightarrow\mathbb{R} is an exterior electric potential, and \varepsilon is a small positive number. If A=0 and \varepsilon=\hbar is Planck's constant this problem is equivalent to the Schrödinger-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 isometries of \mathbb{R}^{3}. Then, for any given homomorphism \tau:G\rightarrow\mathbb{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:\mathbb{R} ^{3}\rightarrow\mathbb{C} which satisfy u(gx)=\tau(g)u(x) for all g\in G, x\in\mathbb{R}^{3}. We also study the concentration behavior of these solutions as \varepsilon → 0.