Cingolani, Silvia; Clapp, Mónica; Secchi, Simone Intertwining semiclassical solutions to a Schrödinger-Newton system. (English) Zbl 1260.35198 Discrete Contin. Dyn. Syst., Ser. S 6, No. 4, 891-908 (2013). Summary: We study the problem \[ \begin{cases} (-\varepsilon \mathrm{i} \nabla + A(x))^{2}u + V(x)u = \varepsilon^{-2} \left( \frac{1}{|x|} \ast |u|^{2} \right) u, \\ u \in L^{2}(\mathbb{R}^{3}, \mathbb{C}), \varepsilon \nabla u + \mathrm{i} Au \in L^{2}(\mathbb{R}^{3}, \mathbb{C}^{3}), \end{cases} \] 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 R. Penrose [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 356, No. 1743, 1927–1939 (1998; Zbl 1152.81659)] 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 \(\tau \colon G \rightarrow \mathbb{S}^{1}\) 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 \(u\colon \mathbb{R}^{3} \rightarrow \mathbb{C}\) 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 \(\varepsilon \rightarrow 0\). Cited in 1 ReviewCited in 29 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q40 PDEs in connection with quantum mechanics 35J20 Variational methods for second-order elliptic equations 35B06 Symmetries, invariants, etc. in context of PDEs Keywords:Schrödinger-Newton system; nonlocal nonlinearity; electromagnetic potential; semiclassical solutions; intertwining solutions Citations:Zbl 1152.81659 PDFBibTeX XMLCite \textit{S. Cingolani} et al., Discrete Contin. Dyn. Syst., Ser. S 6, No. 4, 891--908 (2013; Zbl 1260.35198) Full Text: DOI arXiv