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Schrödinger Operator on the Zigzag Half-Nanotube in Magnetic Field

A. Iantchenko, E. Korotyaev (2010)

Mathematical Modelling of Natural Phenomena

We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic field which is described by the magnetic Schrödinger operator with a periodic potential plus a finitely supported perturbation. We describe all eigenvalues and resonances of this operator, and theirs dependence on the magnetic field. The proof is reduced to the analysis of the periodic Jacobi operators on the half-line with finitely supported perturbations.

Seeable matter; unseeable antimatter

Geoffrey Dixon (2014)

Commentationes Mathematicae Universitatis Carolinae

The universe we see gives every sign of being composed of matter. This is considered a major unsolved problem in theoretical physics. Using the mathematical modeling based on the algebra 𝐓 : = 𝐂 ⊗ 𝐇 ⊗ 𝐎 , an interpretation is developed that suggests that this seeable universe is not the whole universe; there is an unseeable part of the universe composed of antimatter galaxies and stuff, and an extra 6 dimensions of space (also unseeable) linking the matter side to the antimatter—at the very least.

Semiclassics of the quantum current in very strong magnetic fields

Soren Fournais (2002)

Annales de l’institut Fourier

We prove a formula for the current in an electron gas in a semiclassical limit corresponding to strong, constant, magnetic fields. Little regularity is assumed for the scalar potential V . In particular, the result can be applied to the mean field from magnetic Thomas-Fermi theory V MTF . The proof is based on an estimate on the density of states in the second Landau band.

Sharp trace asymptotics for a class of 2 D -magnetic operators

Horia D. Cornean, Søren Fournais, Rupert L. Frank, Bernard Helffer (2013)

Annales de l’institut Fourier

In this paper we prove a two-term asymptotic formula for the spectral counting function for a 2 D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2 D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure...

Soliton-pair Propagation under Thermal Bath Effect

N. Boutabba, H. Eleuch (2012)

Mathematical Modelling of Natural Phenomena

We consider two atomic transitions excited by two variable laser fields in a three-level system. We study the soliton-pair propagation out of resonance and under thermal bath effect. We present general analytical implicit expression of the soliton-pair shape. Furthermore, we show that when the coupling to the environment exceeds a critical value, the soliton-pair propagation through three-level atomic system will be prohibited.

Solitons and Gibbs Measures for Nonlinear Schrödinger Equations

K. Kirkpatrick (2012)

Mathematical Modelling of Natural Phenomena

We review some recent results concerning Gibbs measures for nonlinear Schrödinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.

Solutions of the Dirac-Fock equations without projector

Éric Paturel (2000)

Journées équations aux dérivées partielles

In this paper we prove the existence of infinitely many solutions of the Dirac-Fock equations with N electrons turning around a nucleus of atomic charge Z , satisfying N < Z + 1 and α max ( Z , N ) < 2 / ( 2 / π + π / 2 ) , where α is the fundamental constant of the electromagnetic interaction (approximately 1/137). This work is an improvement of an article of Esteban-Séré, where the same result was proved under more restrictive assumptions on N .

Sparse grids for the Schrödinger equation

Michael Griebel, Jan Hamaekers (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore...

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