Electron. J. Diff. Eqns., Vol. 2004(2004), No. 123, pp. 19.
Nonlinear subelliptic Schrodinger equations with
external magnetic field
Kyril Tintarev
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
in the subelliptic symbol
by
,
where
is called a
magnetic potential for the magnetic field
.
We prove existence of ground state solutions (Sobolev minimizers)
for nonlinear Schrodinger equation associated with such
Hamiltonian on a generally, noncompact Riemannian manifold,
generalizing the existence result of EstebanLions [5]
for the nonlinear Schrödinger equation with
a constant magnetic field on
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 LaplaceBeltrami 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
email: kyril.tintarev@math.uu.se 
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