Nonlinear subelliptic Schrödinger equations with external magnetic field

Tintarev, Kyril

Electronic journal of differential equations, Tome 2004 (2004)

Résumé

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 the momentum operator $\frac 1i d$ in the subelliptic symbol by $\frac 1i d-\alpha$, where $\alpha\in TM^*$ is called a magnetic potential for the magnetic field $\beta=d\alpha$. We prove existence of ground state solutions (Sobolev minimizers) for nonlinear Schrodinger 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 $\mathbb{R}^N$ 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.

Détail

Classification : 35H20, 35J60, 35Q60, 43A85, 58J05
Keywords: homogeneous spaces, magnetic field, Schrödinger operator, subelliptic operators, semilinear equations, weak convergence, concentration compactness