Nonlinear subelliptic Schrodinger equations with external magnetic field
Date
2004-10-18
Authors
Tintarev, Kyril
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
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 momentum operator 1/i d in the subelliptic symbol by 1/i d - α, where α ∈ TM* is called a magnetic potential for the magnetic field β =dα.
We prove existence of ground state solutions (Sobolev minimizers) for non-linear Schrödinger 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 ℝN 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.
Description
Keywords
Homogeneous spaces, Magnetic field, Schrödinger operator, Subelliptic operators, Semilinear equations, Weak convergence, Concentration compactness
Citation
Tintarev, K. (2004). Nonlinear subelliptic Schrodinger equations with external magnetic field. <i>Electronic Journal of Differential Equations, 2004</i>(123), pp. 1-9.
Rights
Attribution 4.0 International