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Eight exactly solvable complex potentials in Bender-Boettcher quantum mechanics

Znojil, Miloslav (2001)

Proceedings of the 20th Winter School "Geometry and Physics"

This is a readable review of recent work on non-Hermitian bound state problems with complex potentials. A particular example is the generalization of the harmonic oscillator with the potentials: V ( x ) = ω 2 2 x - 2 i β ω 2 - ω 2 . Other examples include complex generalizations of the Morse potential, the spiked radial harmonic potential, the Kratzer-Coulomb potential, the Rosen Morse oscillator and others. Instead of demanding Hermiticity H = H * the condition required is H = P T H P T where P changes the parity and T transforms i to - i .

Electronic properties of disclinated nanostructured cylinders

R. Pincak, J. Smotlacha, M. Pudlak (2013)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

The electronic structure of the nanocylinder is investigated. Two cases of this kind of the nanostructure are explored: the defect-free nanocylinder and the nanocylinder whose geometry is perturbed by 2 heptagonal defects lying on the opposite sides. The characteristic quantity which is of our interest is the local density of states. To calculate it, the continuum gauge field-theory model will be used. In this model, the Dirac-like equation is solved on a curved surface. This procedure was used...

Elementary linear algebra for advanced spectral problems

Johannes Sjöstrand, Maciej Zworski (2007)

Annales de l’institut Fourier

We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical...

Embedded eigenvalues and resonances of Schrödinger operators with two channels

Xue Ping Wang (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

In this article, we give a necessary and sufficient condition in the perturbation regime on the existence of eigenvalues embedded between two thresholds. For an eigenvalue of the unperturbed operator embedded at a threshold, we prove that it can produce both discrete eigenvalues and resonances. The locations of the eigenvalues and resonances are given.

Energy machineries on a manifold; application to the construction of new energy representations of Gauge groups.

Jean-Yves Marion (1990)

Publicacions Matemàtiques

The introduction of the concepts of energy machinery and energy structure on a manifold makes it possible a large class of energy representations of gauge groups including, as a very particular case, the ones known up to now. By using an adaptation of methods initiated by I. M. Gelfand, we provide a sufficient condition for the irreducibility of these representations.

Entropy generation in a model of reversible computation

Diego de Falco, Dario Tamascelli (2006)

RAIRO - Theoretical Informatics and Applications

We present a model in which, due to the quantum nature of the signals controlling the implementation time of successive unitary computational steps, physical irreversibility appears in the execution of a logically reversible computation.

Entwining Yang-Baxter maps and integrable lattices

Theodoros E. Kouloukas, Vassilios G. Papageorgiou (2011)

Banach Center Publications

Yang-Baxter (YB) map systems (or set-theoretic analogs of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L₁, L₂, L₃ derived from symplectic leaves of 2 × 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L₁ = L₂ = L₃ this property yields the set-theoretic quantum Yang-Baxter equation, i.e. the...

Équations de champ moyen pour la dynamique quantique d’un grand nombre de particules

Patrick Gérard (2003/2004)

Séminaire Bourbaki

L’objet de cet exposé est de montrer comment l’évolution de Schrödinger pour le problème à N corps quantique est approchée, lorsque N tend vers l’infini, dans un régime convenable, par une évolution non-linéaire en dimension trois d’espace. On traitera le cas des bosons, qui conduit à l’équation de Schrödinger-Poisson, et celui des fermions, qui débouche sur le système de Hartree-Fock.

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