Objective
The project is devoted to problems lying in the intersection of such areas as spectral theory of PDE's, noncommutative geometry, semiclassical asymptotics and quantization, operator extension theory, and fractal geometry. The realisation of this project will contribute to the discovery of new relations between these areas and to understanding the mathematical reasons for the surprising behaviour of physical systems in a magnetic field.
New approaches in the following fields will be developed :
- use of the Krein resolvent formula for the Laplace-Beltrami operator with a vector potential perturbed by a potential with discrete or fractal support;
- localized states for Schroedinger-type operators with periodic potentials;
- group theoretical magneto-Bloch analysis and operator extension theory for exactly solvable models;
- semiclassical methods for description of spectral data of PDE's corresponding to complex Lagrangian subvarieties and complex vector bundles;
- noncommutative geometry for algebras with nonlinear commutation relations corresponding to Coulomb-like potentials and the related construction of(Bessel type) coherent states;
- quantum averaging and deaveraging procedure.
These approaches will be applied to:
- obtaining explicit and asymptotic formulas for the spectrum (or parts of the spectrum) and for eigen functions, including formulas for the exponential splitting and gap length of Laplace- Beltrami and Schroedinger operators corresponding to classical integrable systems (such as geodesic flows on 2-sheres and tori, periodic Toda lattice), and to certain partially integrable systems (Hydrogen atom and ion in a homogeneous magnetic field), as well as to homogeneous flows with periodic point perturbation;
- finding conditions for band structure, gap-finiteness and for the appearance of the Landau levels in the spectrum of certain higher dimensional Schroedinger-type operators, obtaining there by the "flux-energy" diagram analogous to the "Hofstadter butterfly", calculation of the Chern numbers and Berry phase for these models.
Fields of science (EuroSciVoc)
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Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Call for proposal
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Funding Scheme
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Coordinator
53115 Bonn
Germany
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.