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Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

Michael Melgaard (2003)

Open Mathematics

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H m+Hom+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H m, H om) is established and asymptotic expansions...

Quasiharmonic fields and Beltrami operators

Claudia Capone (2002)

Commentationes Mathematicae Universitatis Carolinae

A quasiharmonic field is a pair ℱ = [ B , E ] of vector fields satisfying div B = 0 , curl E = 0 , and coupled by a distorsion inequality. For a given ℱ , we construct a matrix field 𝒜 = 𝒜 [ B , E ] such that 𝒜 E = B . This remark in particular shows that the theory of quasiharmonic fields is equivalent (at least locally) to that of elliptic PDEs. Here we stress some properties of our operator 𝒜 [ B , E ] and find their applications to the study of regularity of solutions to elliptic PDEs, and to some questions of G-convergence.

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