Intertwining semiclassical solutions to a Schrödinger-Newton system

2011/10/19 by Silvia Cingolani, Cingolani, Silvia, Mónica Clapp +3
Mathematics · Physics and Astronomy · #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1110.4213

openalex publication_date 2011/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the problem (-εi∇+A(x)) ²u+V(x)u=ε^-2((1)/(|x|)∗|u|²) u, u∈ L²(ℝ³,ℂ), \ε∇ u+iAu∈ L²(ℝ³,ℂ³), where A\colonℝ³→ℝ³ is an exterior magnetic potential, V\colonℝ³→ℝ is an exterior electric potential, and ε is a small positive number. If A=0 and ε=ℏ is Planck's constant this problem is equivalent to the Schrödinger-Newton equations proposed by Penrose in \citepe2 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 ℝ³. Then, for any given homomorphism τ:G→\mathbbS¹ 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)=τ(g)u(x) for all g∈ G, x∈ℝ³. We also study the concentration behavior of these solutions as ε→0.\medskip

Citations

• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• On Gravity's role in Quantum State Reduction

Related

• Concentration phenomena for the Schr "odinger-Poisson system in\n \ℝ²
• Semiclassical states for coupled nonlinear Schrödinger equations with a critical frequency
• Infinitely many segregated vector solutions of Schrodinger system
• Semiclassical states for the curl-curl problem
• Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system