DSpace Collection: Volume 34, Number 1
http://hdl.handle.net/10525/2575
Serdica Mathematical Journal Volume 34, Number 1, 2008The Collection's search engineSearch the Channelsearch
http://sci-gems.math.bas.bg/jspui/simple-search
Generalized Backscattering and the Lax-Phillips Transform
http://hdl.handle.net/10525/2593
Title: Generalized Backscattering and the Lax-Phillips Transform<br/><br/>Authors: Melrose, Richard; Uhlmann, Gunther<br/><br/>Abstract: Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle w = Sq in terms of the incoming angle with S orthogonal and Id-S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the modified backscattering map is a local isomorphism near elements of a generic set of potentials.<br/><br/>Description: 2000 Mathematics Subject Classification: 35P25, 35R30, 58J50.Supersymmetry and Ghosts in Quantum Mechanics
http://hdl.handle.net/10525/2592
Title: Supersymmetry and Ghosts in Quantum Mechanics<br/><br/>Authors: Robert, Didier<br/><br/>Abstract: A standard supersymmetric quantum system is defined by a Hamiltonian [^H] = ½([^Q]*[^Q] +[^Q][^Q]*), where the super-charge [^Q] satisfies [^Q]2 = 0, [^Q] commutes with [^H]. So we have [^H] ≥ 0 and the quantum spectrum of [^H] is non negative. On the other hand Pais-Ulhenbeck proposed in 1950 a model in quantum-field theory where the d'Alembert operator [¯] = [(∂2)/( ∂t2)] − Δx is replaced by fourth order operator [¯]([¯] + m2), in order to eliminate the divergences occuring in quantum field theory.But then the Hamiltonian of the system, obtained by second quantization, has large negative energies called "ghosts" by physicists. We report here on a joint work with A. Smilga (SUBATECH, Nantes) where we consider a similar problem for some models in quantum mechanics which are invariant under supersymmetric transformations. We show in particular that "ghosts" are still present.<br/><br/>Description: 2000 Mathematics Subject Classification: 81Q60, 35Q40.On an ODE Relevant for the General Theory of the Hyperbolic Cauchy Problem
http://hdl.handle.net/10525/2591
Title: On an ODE Relevant for the General Theory of the Hyperbolic Cauchy Problem<br/><br/>Authors: Bernardi, Enrico; Bove, Antonio<br/><br/>Abstract: In this paper we study an ODE in the complex plane. This is a key step in the search of new necessary conditions for the well posedness of the Cauchy Problem for hyperbolic operators with double characteristics.<br/><br/>Description: 2000 Mathematics Subject Classification: 34E20, 35L80, 35L15.Resolvent and Scattering Matrix at the Maximum of the Potential
http://hdl.handle.net/10525/2590
Title: Resolvent and Scattering Matrix at the Maximum of the Potential<br/><br/>Authors: Alexandrova, Ivana; Bony, Jean-François; Ramond, Thierry<br/><br/>Abstract: We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing the structure of the spectral function and the scattering matrix of the Schrödinger operator at the critical energy.<br/><br/>Description: 2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38.Spectral Shift Function for the Perturbations of Schrödinger Operators at High Energy
http://hdl.handle.net/10525/2589
Title: Spectral Shift Function for the Perturbations of Schrödinger Operators at High Energy<br/><br/>Authors: Assel, Rachid; Dimassi, Mouez<br/><br/>Abstract: We give a complete pointwise asymptotic expansion for the Spectral Shift Function for Schrödinger operators that are perturbations of the Laplacian on Rn with slowly decaying potentials.<br/><br/>Description: 2000 Mathematics Subject Classification: 35P20, 35J10, 35Q40.Spectra of Ruelle Transfer Operators for Contact Flows
http://hdl.handle.net/10525/2588
Title: Spectra of Ruelle Transfer Operators for Contact Flows<br/><br/>Authors: Stoyanov, Luchezar<br/><br/>Abstract: In this survey article we discuss some recent results concerning strong spectral estimates for Ruelle transfer operators for contact flows on basic sets similar to these of Dolgopyat obtained in the case of Anosov flows with C1 stable and unstable foliations. Some applications of Dolgopyat's results and the more recent ones are also described.Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator
http://hdl.handle.net/10525/2587
Title: Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator<br/><br/>Authors: Astaburuaga, María Angélica; Briet, Philippe; Bruneau, Vincent; Fernández, Claudio; Raikov, Georgi<br/><br/>Abstract: We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant ϰ of the perturbation tends to zero. Further, under the assumption that the Fermi Golden Rule holds true, we deduce estimates for the time evolution of the resonance states with and without analyticity assumptions; in the second case we obtain these results as a corollary of suitable Mourre estimates and a recent article of Cattaneo, Graf and Hunziker [11]. Next, we describe sets of perturbations V for which the Fermi Golden Rule is valid at each embedded eigenvalue of H; these sets turn out to be dense in various suitable topologies. Finally, we assume that V decays fast enough at infinity and is of definite sign, introduce the Krein spectral shift function for the operator pair (H+V, H), and study its singularities at the energies which coincide with eigenvalues of infinite multiplicity of the unperturbed operator H.On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness
http://hdl.handle.net/10525/2586
Title: On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness<br/><br/>Authors: Nishitani, Tatsuo<br/><br/>Abstract: In this paper we prove that for non effectively hyperbolic operators with smooth double characteristics with the Hamilton map exhibitinga Jordan block of size 4 on the double characteristic manifold the Cauchyproblem is well posed in the Gevrey 6 class if the strict Ivrii-Petkov-Hörmander condition is satisfied.<br/><br/>Description: 2000 Mathematics Subject Classification: 35L15, Secondary 35L30.Global Waves with Non-Positive Energy in General Relativity
http://hdl.handle.net/10525/2585
Title: Global Waves with Non-Positive Energy in General Relativity<br/><br/>Authors: Bachelot, Alain<br/><br/>Abstract: The theory of the waves equations has a long history since M.Riesz and J. Hadamard. It is impossible to cite all the important results inthe area, but we mention the authors related with our work: J. Leray [34]and Y. Choquet-Bruhat [9] (Cauchy problem), P. Lax and R. Phillips [33](scattering theory for a compactly supported perturbation), L. H¨ ormander[27] and J-M. Bony [7] (microlocal analysis). In all these domains, V. Petkovhas made fundamental contributions, mainly in microlocal analysis, scattering theory, dynamical zeta functions (see in particular the monography [42]).In this paper we present a survey of some recent results on the globalexistence and the asymptotic behaviour of waves, when the conserved energyis not definite positive. This unusual situation arises in important cosmological models of the General Relativity where the gravitational curvatureis very strong. We consider the case of the closed time-like curves (violationof the causality) [1], and the charged black-holes (superradiance) [3]<br/><br/>Description: 2000 Mathematics Subject Classification: 35Lxx, 35Pxx, 81Uxx, 83Cxx.Accurate WKB Approximation for a 1D Problem with Low Regularity
http://hdl.handle.net/10525/2584
Title: Accurate WKB Approximation for a 1D Problem with Low Regularity<br/><br/>Authors: Nier, F.<br/><br/>Abstract: This article is concerned with the analysis of the WKB expansion in a classically forbidden region for a one dimensional boundary valueSchrodinger equation with a non smooth potential. The assumed regularityof the potential is the one coming from a non linear problem and seems to bethe critical one for which a good exponential decay estimate can be provedfor the first remainder term. The treatment of the boundary conditionsbrings also some interesting subtleties which require a careful application ofCarleman’s method.<br/><br/>Description: 2000 Mathematics Subject Classification: 34L40, 65L10, 65Z05, 81Q20.Microlocal Approach to Tensor Tomography and Boundary and Lens Rigidity
http://hdl.handle.net/10525/2583
Title: Microlocal Approach to Tensor Tomography and Boundary and Lens Rigidity<br/><br/>Authors: Stefanov, Plamen<br/><br/>Abstract: This is a survey of the recent results by the author and GuntherUhlmann on the boundary rigidity problem and on the associated tensortomography problem.<br/><br/>Description: 2000 Mathematics Subject Classification: 53C24, 53C65, 53C21.A Simple Example of Localized Parametric Resonance for the Wave Equation
http://hdl.handle.net/10525/2582
Title: A Simple Example of Localized Parametric Resonance for the Wave Equation<br/><br/>Authors: Colombini, Ferruccio; Rauch, Jeffrey<br/><br/>Abstract: The problem studied here was suggested to us by V. Petkov.Since the beginning of our careers, we have benefitted from his insights inpartial differential equations and mathematical physics. In his writings andmany discussions, the conjuction of deep analysis and specially interestingproblems has been a source inspiration for us.<br/><br/>Description: 2000 Mathematics Subject Classification: 35L05, 35P25, 47A40.Weighted Dispersive Estimates for Solutions of the Schrödinger Equation
http://hdl.handle.net/10525/2581
Title: Weighted Dispersive Estimates for Solutions of the Schrödinger Equation<br/><br/>Authors: Cardoso, Fernando; Cuevas, Claudio; Vodev, Georgi<br/><br/>Abstract: Introduction and statement of results. In the present paper we will be interested in studying the decay properties of the Schrödinger group.<br/><br/>Description: 2000 Mathematics Subject Classification: 35L15, 35B40, 47F05.Pseudodifferential Operators and Weighted Normed Symbol Spaces
http://hdl.handle.net/10525/2580
Title: Pseudodifferential Operators and Weighted Normed Symbol Spaces<br/><br/>Authors: Sjöstrand, J.<br/><br/>Abstract: This work is the continuation of two earlier ones by the author and stimulated by many more recent contributions. We develop a verygeneral calculus of pseudodifferential operators with microlocally definednormed symbol spaces. The goal was to attain the natural degree of generality in the case when the underlying metric on the cotangent space isconstant. We also give sufficient conditions for our operators to belong toSchatten–von Neumann classes.<br/><br/>Description: 2000 Mathematics Subject Classification: 35S05.