Nonlinear subelliptic Schrodinger equations with external magnetic field




Tintarev, Kyril

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Texas State University-San Marcos, Department of Mathematics


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.



Homogeneous spaces, Magnetic field, Schrödinger operator, Subelliptic operators, Semilinear equations, Weak convergence, Concentration compactness


Tintarev, K. (2004). Nonlinear subelliptic Schrodinger equations with external magnetic field. <i>Electronic Journal of Differential Equations, 2004</i>(123), pp. 1-9.


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