Hamiltonian dynamics and spectral theory for spin-oscillators

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.Comment: 32 page

ИССЛЕДОВАНИЕ ВОЗМОЖНОСТИ КООРДИНАЦИИ ПРОЦЕССОВ СЪЕМКИ ПОВЕРХНОСТИ ЗЕМЛИ КОСМИЧЕСКИМИ АППАРАТАМИ VNREDSat-1 (ВЬЕТНАМ) И БКА (БЕЛАРУСЬ)

Leading companies in the field of obtaining data from Earth remote sensing from space are building Satellite constellations to increase the frequency (multiplicity) and monitor various parts of the Earth’s surface. The Satellite constellation remote sensing equipment from Airbus (France) allows daily observation of any point on the Earth’s surface with high spatial resolution, and the grouping of DigitalGlobe (USA) – with ultra-high spatial resolution. The analysis of the mutual exploitation of the national space systems of the Socialist Republic of Vietnam and the Republic of Belarus has been carried out. There have been analyzed the joint capabilities of VNREDSat-1 (Vietnam) and BKA (Belarus) Earth Observation Systems for the monitoring of natural resources, the environment and natural disasters by comparison of their space and temporal characteristics (orbit parameters and re-visit time) as well as spectral-energetic characteristics (space, spectral and radiometric resolutions). The accepted correlation of technical parameters for solving the previously mentioned tasks having into account the different phases of the mentioned satellites has been determined. The conclusions have been reinforced by the results of live experiments with archival and instant data from both EO satellites. Ведущие компании в области получения данных дистанционного зондирования Земли из космоса строят группировки космических аппаратов для увеличения периодичности (кратности) и ведения мониторинга различных участков земной поверхности. Группировка космических аппаратов дистанционного зондирования компании Airbus (Франция) позволяет ежедневно наблюдать любую точку поверхности Земли с высоким пространственным разрешением, а группировка компании DigitalGlobe (США) – со сверхвысоким пространственным разрешением. Наряду с этим запускается большое количество аппаратов дистанционного зондирования Земли в рамках национальных космических программ стран. Группировки космических аппаратов в таких странах незначительные. Проведен анализ перспектив совместного использования национальных космических систем Социалистической Республики Вьетнам и Республики Беларусь. Путем сопоставления пространственно-временных (орбитальные характеристики, цикличность орбит) и спектрально-энергетических (пространственное, спектральное и радиометрическое разрешение) характеристик функционирующих в настоящее время космических аппаратов дистанционного зондирования Земли VNREDSat-1 (Вьетнам) и БКА (Беларусь) проанализированы возможности их совместного использования для мониторинга природных ресурсов, окружающей среды и стихийных бедствий. Выявлена приемлемая для решения перечисленных задач степень корреляции технических параметров с учетом различных фаз данных космических аппаратов. Выводы подкреплены результатами натурных экспериментов с использованием архивных и оперативных данных дистанционного зондирования Земли, получаемых национальными космическими аппаратами

The affine invariant of generalized semitoric systems

29 pages, 15 figuresInternational audienceA generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which correspond to fibers of F which are topologically multipinched tori. The image F(M) is a singular affine manifold which contains a distinguished set of isolated points in its interior: the focus-focus values {(x_i,y_i)} of F. By performing a vertical cutting procedure along the lines {x:=x_i}, we construct a homeomorphism f : F(M) --> f(F(M)), which restricts to an affine diffeomorphism away from these vertical lines, and generalizes a construction of Vu Ngoc. The set \Delta:=f(F(M)) in R^2 is a symplectic invariant of (M,\omega,F), which encodes the affine structure of F. Moreover, \Delta may be described as a countable union of planar regions of four distinct types, where each type is defined as the region bounded between the graphs of two functions with various properties (piecewise linear, continuous, convex, etc). If F is a toric system, \Delta is a convex polygon (as proven by Atiyah and Guillemin-Sternberg) and f is the identity

Superintegrable Hamiltonian systems: Geometry and perturbations

Many and important integrable Hamiltonian systems are 'superintegrable', in the sense that there is an open subset of their 2d-dimensional phase space in which all motions are linear on tori of dimension n < d. A thorough comprehension of these systems requires a description which goes beyond the standard notion of Liouville - Arnold integrability, that is, the existence of an invariant fibration by Lagrangian tori. Instead, the natural object to look at is formed by both the fibration by the ( isotropic) invariant tori and by its (coisotropic) polar foliation, which together form what in symplectic geometry is called a 'dual pair', or 'bifoliation', or 'bifibration'. We review this geometric structure, relating it to the dynamical properties of superintegrable systems and pointing out its importance for a thorough understanding of these systems