Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 123, pp. 1-9.
Title: Nonlinear subelliptic Schrodinger equations with
external magnetic field
Author: Kyril Tintarev (Uppsala Univ., Sweden)
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 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 Schr\"odinger 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.