%0 Journal Article
%T Nonlinear subelliptic Schrodinger equations with external magnetic field
%A Kyril Tintarev
%J Electronic Journal of Differential Equations
%D 2004
%I Texas State University
%X 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 $alphain TM^*$ is called a magnetic potential for the magnetic field $eta=dalpha$. 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 Schr dinger 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.
%K Homogeneous spaces
%K magnetic field
%K Schrodinger operator
%K subelliptic operators
%K semilinear equations
%K weak convergence
%K concentration compactness.
%U http://ejde.math.txstate.edu/Volumes/2004/123/abstr.html