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Quantum Dynamics and generalized fractal dimensions: an introduction

François Germinet (2002/2003)

Séminaire Équations aux dérivées partielles

We review some recent results on quantum motion analysis, and in particular lower bounds for moments in quantum dynamics. The goal of the present exposition is to stress the role played by quantities we shall call Transport Integrals and by the so called generalized dimensions of the spectral measure in the analysis of quantum motion. We start with very simple derivations that illustrate how these quantities naturally enter the game. Then, gradually, we present successive improvements, up to most...

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...

Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space

Alexander Chebotarev, Dmitry Victorov (1998)

Banach Center Publications

By using F. A. Berezin's canonical transformation method [5], we derive a nonadapted quantum stochastic differential equation (QSDE) as an equation for the strong limit of the family of unitary groups satisfying the Schrödinger equation with singularly degenerating Hamiltonians in Fock space. Stochastic differentials of QSDE generate a nonadapted associative Ito multiplication table, and the coefficients of these differentials satisfy the formal unitarity conditions of the Hudson-Parthasarathy type...

Reduced and extended weak coupling limit

Jan Dereziński, Wojciech De Roeck (2007)

Banach Center Publications

The main aim of our lectures is to give a pedagogical introduction to various mathematical formalisms used to describe open quantum systems: completely positive semigroups, dilations of semigroups, quantum Langevin dynamics and the so-called Pauli-Fierz Hamiltonians. We explain two kinds of the weak coupling limit. Both of them show that Hamiltonian dynamics of a small quantum system interacting with a large resevoir can be approximated by simpler dynamics. The better known reduced weak coupling...

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.

Some results about dissipativity of Kolmogorov operators

Giuseppe Da Prato, Luciano Tubaro (2001)

Czechoslovak Mathematical Journal

Given a Hilbert space H with a Borel probability measure ν , we prove the m -dissipativity in L 1 ( H , ν ) of a Kolmogorov operator K that is a perturbation, not necessarily of gradient type, of an Ornstein-Uhlenbeck operator.

Some spectral properties of the streaming operator with general boundary conditions

Mohamed Boulanouar (2008)

Applications of Mathematics

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K . We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C 0 -semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case K 1 . We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the...

Spectral theory of corrugated surfaces

Vojkan Jakšić (2001)

Journées équations aux dérivées partielles

We discuss spectral and scattering theory of the discrete laplacian limited to a half-space. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics.

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