Nonlinear subelliptic Schrodinger equations with external magnetic field

Abstract

To account for an external magnetic field in a Hamiltonian of a quantum system on a manifold (modelled here by a subelliptic Dirichlet form), one replaces the momentum operator 1/i d in the subelliptic symbol by 1/i d - α, where α ∈ TM* is called a magnetic potential for the magnetic field β =dα.

We prove existence of ground state solutions (Sobolev minimizers) for non-linear Schrödinger equation associated with such Hamiltonian on a generally, non-compact Riemannian manifold, generalizing the existence result of Esteban-Lions [5] for the nonlinear Schrödinger equation with a constant magnetic field on ℝN and the existence result of [6] for a similar problem on manifolds without a magnetic field. The counterpart of a constant magnetic field is the magnetic field, invariant with respect to a subgroup of isometries. As an example to the general statement we calculate the invariant magnetic fields in the Hamiltonians associated with the Kohn Laplacian and for the Laplace-Beltrami operator on the Heisenberg group.