Intertwining semiclassical solutions to a Schrödinger-Newton system

Abstract

We study the problem (-ε\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=ε^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}ε\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ is an exterior magnetic potential, $V:{\mathbb{R}}^3\to \mathbb{R}$ is an exterior electric potential, and $\epsilon$ is a small positive number. If A=0 and $\epsilon =\hslash$ is Planck's constant this problem is equivalent to the Schrödinger-Newton equations proposed by Penrose in \cite{pe2}\ 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 $\tau :G\to {\mathbb{S}}^1$ 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 $u:\mathbb{R}$ 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 $ε\rightarrow0.\medskip$

18 pages, to appear on DCDS-S