[Paper Review] Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities

Silvia Cingolani, Simone Secchi+1 more|ArXiv.org|Mar 31, 2009

Advanced Mathematical Physics Problems21 references20 citations

TL;DR

This paper establishes the existence of multipeak semiclassical solutions to a nonlinear Schrödinger equation with magnetic fields and Hartree-type nonlinearity, showing that solutions concentrate around minimum points of the electric potential as the Planck constant tends to zero. Using a penalization method and variational techniques, it proves that multiple concentration regions form near non-degenerate minima of the potential, even in the presence of nonlocal interactions and magnetic fields.

ABSTRACT

The semi-classical regime of standing wave solutions of a Schrödinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.

Motivation & Objective

• To analyze the semiclassical limit of standing wave solutions for a Schrödinger equation with magnetic potentials and Hartree-type nonlinearity.
• To establish the existence of multipeak solutions that concentrate near the minimum points of the electric potential as the semiclassical parameter tends to zero.
• To extend previous results on nonlocal nonlinearities to systems involving magnetic fields, which had not been addressed in the literature before.
• To prove concentration phenomena in the presence of both nonlocal interactions and electromagnetic fields using variational and penalization techniques.

Proposed method

• Employ a penalization technique recently developed in [13] to construct solutions in a constrained variational framework.
• Use a change of variables to transform the original equation into a form where the semiclassical parameter ε appears in the rescaled potentials and magnetic fields.
• Apply the Pohozaev identity and orbital stability properties to analyze the structure of ground states and their compactness.
• Establish the Palais-Smale condition for the penalized functional to ensure convergence of minimizing sequences.
• Use Moser iteration and bootstrap arguments to prove boundedness of the L∞ norm of solutions.
• Apply comparison principles and decay estimates via Kato's inequality to show exponential decay of solutions away from concentration regions.

Experimental results

Research questions

• RQ1 Do multipeak solutions exist for the semiclassical Schrödinger equation with magnetic fields and Hartree-type nonlinearity?
• RQ2 Where do the concentration regions of the solutions form in the semiclassical limit?
• RQ3 How does the presence of a magnetic potential affect the concentration behavior compared to the case with zero magnetic field?
• RQ4 Can the penalization method be successfully adapted to handle nonlocal Hartree-type nonlinearities in the presence of magnetic fields?
• RQ5 What is the asymptotic behavior of solutions as the semiclassical parameter ε → 0?

Key findings

• A family of solutions exists that concentrate at multiple points near the minimum points of the electric potential as ε → 0.
• The solutions exhibit exponential decay away from the concentration regions, with bounds of the form |uε(x)| ≤ C exp(−c dist(x, (M²β)ε ∪ (Zβ)ε)) for some C, c > 0.
• When the zero set Z of the magnetic potential is nonempty, the solution decays exponentially on (Z²δ)ε, with |uε(y)| ≤ C exp(−c/ε) for some C, c > 0.
• The penalization method successfully yields solutions satisfying the original equation for sufficiently small ε > 0.
• The concentration behavior is robust under the combined effects of nonlocal Hartree-type interactions and magnetic fields.
• The energy functional satisfies the Palais-Smale condition, ensuring convergence of minimizing sequences in the variational framework.