By Philippe Briet, Francois Germinet, Georgi Raikov
This quantity includes the complaints of the convention on Spectral and Scattering thought for Quantum Magnetic structures, which came about at CIRM, Luminy, France, in July 2008. the most goal of this convention used to be to assemble a couple of experts within the mathematical modelling of magnetic phenomena in quantum mechanics, to mark the hot development in addition to to stipulate the longer term improvement during this zone. This quantity comprises unique effects offered by means of a few of the invited audio system and surveys on fresh advances within the mathematical idea of quantum magnetic Hamiltonians. many of the talks on the convention, in addition to the articles during this quantity, were devoted to one of many following issues: Spectral and scattering concept for magnetic Schrodinger operators; Magnetic Pauli and Dirac operators; Magnetic operators on manifolds; Microlocal research of magnetic Hamiltonians; Random Schrodinger operators and quantum corridor influence; Ginsburg - Landau equation, supraconductivity, magnetic bottles; Bose - Einstein condensate, Gross - Pitaevski equation; and, Magnetic Lieb - Thirring inequalities, balance of subject
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Additional info for Spectral and Scattering Theory for Quantum Magnetic Systems: July 7-11, 2008 Cirm, Luminy Marseilles, France
Sample text
In the second case we obtain that the singular continuous spectrum of H is empty, and any compact subset of the complement of the thresholds set contains at most a finite number of eigenvalues of H, each of them having finite multiplicity. Moreover these Mourre estimates together with some of their spectral consequences generalize to the case of 2D magnetic Schr¨ odinger operators defined on R2 for suitable confining potentials modeling Dirichlet boundary conditions. 1. 1. Edge currents and positive commutator estimates.
GUILLOT 20 [c] = 1 σ − g(Aσj,τ (x , 0) − Aσj,τ (0)) + gRe(hj,τ (x ) − hj,τ (0), gτ,E ) σ m j=1,2 × pj − eaj (x ) − gAj,σ (x , 0) + gRe(hj,σ (x ), gσ,Eσ ) σ ), − gEσ [Aσ3,τ (x , 0) − Aσ3,τ (0, 0)] + gEσ Re(h3,τ (x ) − h3,τ (0), gτ,E σ 1 σ σ σ [d] = gEσ (h3,τ (0), gτ,E )− Eσ (k3 gτ,E , gτ,E ), σ σ σ 2 1 1 σ σ σ σ ) − (k3 gτ,E , gτ,E ) − gAσ3,τ (x , 0) + gRe(h3,τ (x ), gτ,E ) − Φ(k3 gτ,E [e] = σ σ σ σ 2 2 1 ren ren × ∇Hσ,E − Eσ + − Eσ ∇Hσ,E σ σ 2 1 σ σ σ σ ) − (k3 gτ,E , gτ,E ) − gAσ3,τ (x , 0) + gRe(h3,τ (x ), gτ,E ) .
149 (1998), 193-210. [7] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. , 78, 391-408 (1981). [8] M. Reed, B. 1-4, Academic Press, New York, 1975-1979. cl This page intentionally left blank Contemporary Mathematics Volume 500, 2009 Mourre estimates for a 2D magnetic quantum Hamiltonian on strip-like domains Philippe Briet, Peter D. Hislop, Georgi Raikov, and Eric Soccorsi Abstract. We consider a 2D Schr¨ odinger operator H0 with constant magnetic field defined on a strip of finite width.