Electron. J. Diff. Eqns., Vol. 2004(2004), No. 123, pp. 1-9.

Nonlinear subelliptic Schrodinger equations with external magnetic field

Kyril Tintarev

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.

Submitted July 9, 2004. Published October 18, 2004.
Math Subject Classifications: 35H20, 35J60, 35Q60, 43A85, 58J05.
Key Words: Homogeneous spaces; magnetic field; Schrodinger operator; subelliptic operators; semilinear equations; weak convergence; concentration compactness.

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Kyril Tintarev
Department of Mathematics
Uppsala University
P. O. Box 480
75106 Uppsala, Sweden
e-mail: kyril.tintarev@math.uu.se

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