Dissertations / Theses on the topic 'Microlocal and semiclassical analysis'
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Prouff, Antoine. "Correspondance classique-quantique et application au contrôle d'équations d'ondes et de Schrödinger dans l'espace euclidien." Electronic Thesis or Diss., université Paris-Saclay, 2024. https://theses.hal.science/tel-04634673.
Wave and Schrödinger equations model a variety of phenomena, such as propagation of light, vibrating structures or the time evolution of a quantum particle. In these models, the high-energy asymptotics can be approximated by classical mechanics, as geometric optics. In this thesis, we study several applications of this principle to control problems for wave and Schrödinger equations in the Euclidean space, using microlocal analysis.In the first two chapters, we study the damped wave equation and the Schrödinger equation with a confining potential in the euclidean space. We provide necessary and sufficient conditions for uniform stability in the first case, or observability in the second one. These conditions involve the underlying classical dynamics which consists in a distorted version of geometric optics, due to the presence of the potential.Then in the third part, we analyze the quantum-classical correspondence principle in a general setting that encompasses the two aforementioned problems. We prove a version of Egorov's theorem in the Weyl--Hörmander framework of metrics on the phase space. We provide with various examples of application of this theorem for Schrödinger, half-wave and transport equations
Le, Floch Yohann. "Théorie spectrale inverse pour les opérateurs de Toeplitz 1D." Phd thesis, Université Rennes 1, 2014. http://tel.archives-ouvertes.fr/tel-01065441.
Tarkhanov, Nikolai, and Nikolai Vasilevski. "Microlocal analysis of the Bochner-Martinelli integral." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/3001/.
Schultka, Konrad. "Microlocal analyticity of Feynman integrals." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20161.
We give a rigorous construction of analytically regularized Feynman integrals in D-dimensional Minkowski space as meromorphic distributions in the external momenta, both in the momentum and parametric representation. We show that their pole structure is given by the usual power-counting formula and that their singular support is contained in a microlocal generalization of the alpha-Landau surfaces. As further applications, we give a construction of dimensionally regularized integrals in Minkowski space and prove discontinuity formula for parametric amplitudes.
Ramaseshan, Karthik. "Microlocal analysis of the doppler transform on R³ /." Thesis, Connect to this title online; UW restricted, 2003. http://hdl.handle.net/1773/5739.
Welch, Barry Alan. "Semiclassical analysis of vibroacoustic systems." Thesis, University of Southampton, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.433930.
Webber, James. "Radon transforms and microlocal analysis in Compton scattering tomography." Thesis, University of Manchester, 2018. https://www.research.manchester.ac.uk/portal/en/theses/radon-transforms-and-microlocal-analysis-in-compton-scattering-tomography(c1ad3583-01ce-4147-8576-2e635090cb15).html.
Conrady, Florian. "Semiclassical analysis of loop quantum gravity." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=982087144.
Conrady, Florian. "Semiclassical analysis of loop quantum gravity." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2006. http://dx.doi.org/10.18452/15549.
In this Ph.D. thesis, we explore and develop new methods that should help in determining an effective semiclassical description of canonical loop quantum gravity and spin foam gravity. A brief introduction to loop quantum gravity is followed by three research papers that present the results of the Ph.D. project. In the first article, we deal with the problem of time and a new proposal for implementing proper time as boundary conditions in a sum over histories: we investigate a concrete realization of this formalism for free scalar field theory. In the second article, we translate semiclassical states of linearized gravity into states of loop quantum gravity. The properties of the latter indicate how semiclassicality manifests itself in the loop framework, and how this may be exploited for doing semiclassical expansions. In the third part, we propose a new formulation of spin foam models that is fully triangulation- and background-independent: by means of a symmetry condition, we identify spin foam models whose triangulation-dependence can be naturally removed.
Teloni, Daniele. "Semiclassical analysis of systems of Schrödinger equations." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/19239/.
Di, Gesù Giacomo. "Semiclassical spectral analysis of discrete Witten Laplacians." Phd thesis, Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2013/6528/.
In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten.
Takuwa, Hideki. "Microlocal Analysis of Linear Partial Differential Equations via the FBI Transform." 京都大学 (Kyoto University), 2004. http://hdl.handle.net/2433/148283.
Schubert, Roman. "Semiclassical localization in phase space." Ulm : Universität Ulm, Fakultät für Naturwissenschaften, 2001. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10028611.
Chervova, O. "The massless Dirac equation from the continuum mechanics and microlocal analysis perspectives." Thesis, University College London (University of London), 2012. http://discovery.ucl.ac.uk/1362431/.
Hedeman, Austin J. "Semiclassical Analysis of Fundamental Amplitudes in Loop Quantum Gravity." Thesis, University of California, Berkeley, 2015. http://pqdtopen.proquest.com/#viewpdf?dispub=3686321.
Spin networks arise in many areas of physics and are a key component in both the canonical formulation (loop quantum gravity) and the path-integral formulation (spin-foam gravity) of quantum gravity. In loop quantum gravity the spin networks are used to construct a countable basis for the physical Hilbert space of gravity. The basis states may be interpreted as gauge-invariant wavefunctionals of the connection. Evaluating the wavefunctional on a specific classical connection involves embedding the spin network into a spacelike hypersurface and finding the holonomy around the network. This is equivalent to evaluating a ''g-inserted'' spin network (a spin network with a group action acting on all of the edges of the network). The spin-foam approach to quantum gravity is a path-integral formulation of loop quantum gravity in which the paths are world-histories of embedded spin networks. Depending on the spin-foam model under consideration the vertex amplitude (the contribution a spin-foam vertex makes to the transition amplitude) may be represented by a specific simple closed spin network. The most important examples use the 6j-symbol, the 15j-symbol, and the Riemannian 10j-symbol. The semiclassical treatment of spin networks is the main theme of this dissertation.
To show that classical solutions of general relativity emerge in the appropriate limits of loop quantum gravity or spin-foam gravity requires knowledge of the semiclassical limits of spin networks. This involves interpreting the spin networks as inner products and then treating the inner products semiclassically using the WKB method and the stationary phase approximation. For any given spin network there are many possible inner product models which correspond to how the spin network is ''split up'' into pieces. For example the 6 j-symbol has been studied in both a model involving four angular momenta (Aquilanti et al 2012) and a model involving twelve angular momenta (Roberts 1999). Each of these models offers advantages and disadvantages when performing semiclassical analyses. Since the amplitude of the stationary phase approximation relies on determinants they are easiest to calculate in phase spaces with the fewest dimensions. The phase, on the other hand, is easiest to compute in cases where all angular momenta are treated on an equal footing, requiring a larger phase space.
Surprisingly, the different inner product models are not related by symplectic reduction (the removal of a symmetry from a Hamiltonian system). There is a connection between the models, however. On the level of linear algebra the connection is made by considering first not inner products but matrix elements of linear operators. A given matrix element can then be interpreted as an inner product in two different Hilbert spaces. We call the connection between these two inner product models the ''remodeling of an inner product.'' The semiclassical version of an inner product remodeling is a generalization of the idea that the phase space manifold that supports the semiclassical approximation of a unitary operator may be considered the graph of a symplectomorphism. We use the manifold that supports the semiclassical approximation of the linear map to ''transport'' features from one space to another. Using this transport procedure we can show that the amplitude and phase calculations in the phase spaces for the two models are identical. The asymptotics of a complicated spin network, and thus the fundamental amplitudes of loop quantum gravity and spin-foam gravity, may be computed by first setting up an inner product remodeling and then picking and choosing which features of the calculation to perform in which space.
In this dissertation we first introduce the remodeling of an inner product and the semiclassical features of the remodeling. We then apply the remodeling to the well-studied cases of the 3j-symbol and the 6 j-symbol. Finally we explore how the remodel procedure applies to more complicated spin networks such as the 15j-symbol and the g-inserted spin networks of loop quantum gravity.
Bates, Kenneth A. "Semiclassical analysis of perturbed two-electron states in barium." Columbus, OH : Ohio State University, 2003. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1065906166.
Title from first page of PDF file. Document formatted into pages; contains xii, 170 p.: ill. (some col.). Includes abstract and vita. Advisor: Douglass Schumacher, Dept. of Physics. Includes bibliographical references (p. 125-130).
Tacy, Melissa Evelyn, and melissa tacy@anu edu au. "Semiclassical Lp Estimates for Quasimodes on Submanifolds." The Australian National University. Department of Mathematics, College of Physical & Mathematical Sciences, 2010. http://thesis.anu.edu.au./public/adt-ANU20100622.150105.
Kungsman, Jimmy. "Semiclassical approximation of Dirac resonances using the CAP method." Licentiate thesis, Uppsala universitet, Analys och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-172750.
Peter, Wolfgang [Verfasser]. "Semiclassical analysis of quantum maps with spin orbit coupling / Wolfgang Peter." Ulm : Universität Ulm. Fakultät für Naturwissenschaften, 2013. http://d-nb.info/1030569118/34.
Schulz, René M. [Verfasser], Dorothea [Akademischer Betreuer] Bahns, and Ingo [Akademischer Betreuer] Witt. "Microlocal Analysis of Tempered Distributions / René M. Schulz. Gutachter: Dorothea Bahns ; Ingo Witt. Betreuer: Dorothea Bahns." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2014. http://d-nb.info/1058477277/34.
Di, Gesù Giacomo [Verfasser], and Markus [Akademischer Betreuer] Klein. "Semiclassical spectral analysis of discrete Witten Laplacians / Giacomo Di Gesù. Betreuer: Markus Klein." Potsdam : Universitätsbibliothek der Universität Potsdam, 2013. http://d-nb.info/1035010852/34.
Chocian, Peter. "The semiclassical S-matrix theory of three body Coulomb break-up." Thesis, Royal Holloway, University of London, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.314298.
Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.
Mehl, Craig. "Developing a sorting code for Coulomb excitation data analysis." University of the Western Cape, 2015. http://hdl.handle.net/11394/4871.
This thesis aims at developing a sorting code for Coulomb excitation studies at iThemba LABS. In Coulomb excitation reactions, the inelastic scattering of the projectile transfers energy to the partner nucleus (and vice-versa) through a time-dependent electromagnetic field. At energies well below the Coulomb barrier, the particles interact solely through the well known electromagnetic interaction, thereby excluding nuclear excitations from the process . The data can therefore be analyzed using a semiclassical approximation. The sorting code was used to process and analyze data acquired from the Coulomb excitation of 20Ne beams at 73 and 96 MeV, onto a 194Pt target. The detection of gamma rays was done using the AFRODITE HPGe clover detector array, which consists of nine clover detectors, in coincidence with the 20Ne particles detected with an S3 double-sided silicon detector. The new sorting code includes Doppler-correction effects, charge-sharing, energy and time conditions, kinematics and stopping powers, among others, and can be used for any particle-γ coincidence measurements at iThemba LABS. Results from other Coulomb excitation measurements at iThemba LABS will also be presented.
Krishnan, Venkateswaran P. "A support theorem and an inversion formula for the geodesic ray transform /." Thesis, Connect to this title online; UW restricted, 2007. http://hdl.handle.net/1773/5804.
Pankrachkine, Konstantin. "Semiclassical methods for the two-dimensional Schrödiger operator with a strong magnetic field." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2002. http://dx.doi.org/10.18452/14829.
Spectral properties of the two-dimensional Schroedinger operator with a two-periodic potential and a strong uniform magnetic field is studied with the help of semiclassical methods. The spectral asymptotics is described using the Reeb graph technique. In the case of the rational flux one constructs semiclassical magneto-Bloch functions and describes the asymptotics of the band spectrum on the physical level of proof.
Küster, Benjamin [Verfasser], and Pablo [Akademischer Betreuer] Ramacher. "Semiclassical Analysis of Schrödinger Operators on Closed Manifolds and Symmetry Reduction / Benjamin Küster. Betreuer: Pablo Ramacher." Marburg : Philipps-Universität Marburg, 2015. http://d-nb.info/1081215534/34.
Gutiérrez, Márquez Martha Lucía. "From spectral statistics to decay in quantum chaotic systems : a semiclassical analysis beyond Random Matrix Theory." kostenfrei, 2008. http://www.opus-bayern.de/uni-regensburg/volltexte/2009/1120/.
AMAR-SERVAT, Emmanuelle. "Asymptotic solutions and resonances for Klein-Gordon and Schrödinger operators." Phd thesis, Université Paris-Nord - Paris XIII, 2002. http://tel.archives-ouvertes.fr/tel-00002342.
Ashida, Sohei. "Molecular predissociation resonances below an energy level crossing." Kyoto University, 2018. http://hdl.handle.net/2433/232215.
Lefeuvre, Thibault. "Sur la rigidité des variétés riemanniennes." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS562/document.
A Riemannian manifold is said to be rigid if the length of periodic geodesics (in the case of a closed manifold) or scattered geodesics (in the case of an open manifold) allows to recover the full geometry of the manifold. This notion naturally arises in imaging devices such as X-ray tomography. Thanks to a analytic framework introduced by Guillarmou and based on microlocal analysis (and more precisely on the analytic study of hyperbolic flows of Faure-Sjostrand and Dyatlov-Zworski), we show that the marked length spectrum, that is the lengths of the periodic geodesics marked by homotopy, of a closed Anosov manifold or of an Anosov manifold with hyperbolic cusps locally determines its metric. In the case of an open manifold with hyperbolic trapped set, we show that the length of the scattered geodesics marked by homotopy locally determines the metric. Eventually, in the case of an asymptotically hyperbolic surface, we show that a suitable notion of renormalized distance between pair of points on the boundary at infinity allows to globally reconstruct the geometry of the surface
Nguyen, Duc Tho. "Classical and semi-classical analysis of magnetic fields in two dimensions." Thesis, Rennes 1, 2019. http://www.theses.fr/2019REN1S045/document.
This manuscript is devoted to classical mechanics and quantum mechanics, especially in the presence of magnetic field. In classical mechanics, we use Hamiltonian dynamics to describe the motion of a charged particle in a domain affected by the magnetic field. We are interested in two classical physical problems: the confinement and the scattering problem. In the quantum case, we study the spectral problem of the magnetic Laplacian at the semi-classical level, in two-dimensional domains: on a compact Riemmanian manifold with boundary and on ℝ ². Under the assumption that the magnetic field has a unique positive and non-degenerate minimum, we can describe the eigenfunctions by WKB methods. Thanks to the spectral theorem, we estimated efficiently the true eigenfunctions and the approximate eigenfunctions locally near the minimum point of the magnetic field. On ℝ ², with the additional assumption that the magnetic field is radially symmetric, we can show that the eigenfunctions of the magnetic Laplacian decay exponentially at infinity and at a rate controlled by the phase function created in WKB procedure. Furthermore, the eigenfunctions are very well approximated in an exponentially weighted space
Guidi, Lorenzo [Verfasser], Eric [Akademischer Betreuer] Sonnendrücker, Caroline [Gutachter] Lasser, Eric [Gutachter] Sonnendrücker, and Olivier [Gutachter] Lafitte. "Scattering and reflection of microwave beams in fusion plasmas : Numerical analysis with semiclassical methods / Lorenzo Guidi ; Gutachter: Caroline Lasser, Eric Sonnendrücker, Olivier Lafitte ; Betreuer: Eric Sonnendrücker." München : Universitätsbibliothek der TU München, 2019. http://d-nb.info/1192441826/34.
Bonthonneau, Yannick. "Résonances du laplacien sur les variétés à pointes." Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112141/document.
In this thesis, we study the resonances of the Laplace operator on cusp manifolds. They are manifolds whose ends are real hyperbolic cusps. The resonances were introduced by Selberg in the 50's for the constant curvature cusp surfaces. Their definition was later extended to the case of variable curvature by Lax and Phillips. The resonances are the poles of a meromorphic family of generalized eigenfunctions of the Laplace operator. They are associated to the continuous spectrum of the Laplace operator. To analyze this continuous spectrum, different directions of research are investigated.On the one hand, we obtain results on the localization of resonances. In particular, if the curvature is negative, for a generic set of metrics, they split into two sets. The first one is included in a band near the spectrum. The other is composed of resonances that are far from the spectrum. This leaves a log zone without resonances. On the other hand, we study the microlocal measures associated to certain sequences of spectral parameters. In particular we show that for some sequences of parameters that converge to the spectrum, but not too fast, the associated microlocal measure has to be the Liouville measure. This property holds when the curvature is negative
Kungsman, Jimmy. "Resonances of Dirac Operators." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-223841.
Ingremeau, Maxime. "Ondes planes tordues et diffusion chaotique." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS477/document.
This thesis deals with several problems of scattering theory in the semi-classical limit, that is to say, with properties of the generalised eigenfunctions of a Schrödinger operator at high frequencies. The generalised eigenfunctions of a Schrödinger operator on the Euclidean space, with a compactly supported smooth potential, may always be written as the sum of an incoming wave and an outgoing wave, plus a term which is negligible at infinity. The scattering matrix relates the incoming part with the outgoing part. The first part of this work deals with the spectrum of the scattering matrix. We show an equidistribution result for the eigenvalues of the scattering matrix, under the hypothesis that the sets of fixed points of some maps defined from the classical dynamics has measure zero. This result was previously known under the additional assumption that the classical dynamics has an empty trapped set.A second part of this work deals with the distorted plane waves, which are a particular family of generalized eigenfunctions of a Schrödinger operator, which can be written as the sum of a plane wave and a purely outgoing part. We make the hypothesis that the underlying classical dynamics has a hyperbolic trapped set, and that a certain topological pressure is negative. Under these assumptions, we obtain in the semiclassical limit a precise description of distorted plane waves as a convergent sum of Lagrangian states. In particular, we can deduce from this the semiclassical measure associated to distorted plane waves. If we furthermore assume that the manifold has non-positive curvature, and that the potential is zero, these Lagrangian states project on the base manifold without caustics. We deduce from this results on the C^l norms and on the nodal sets of distorted plane waves. We also obtain a lower bound on the number of nodal domains of the sum of two distorted plane waves with close enough incoming directions , for a small generic perturbation of a metric of negative curvature satisfying the topological pressure assumption
Nakano, Yushi. "Stochastic Stability of Partially Expanding Maps via Spectral Approaches." Kyoto University, 2015. http://hdl.handle.net/2433/200463.
0048
新制・課程博士
博士(人間・環境学)
甲第19200号
人博第741号
新制||人||178(附属図書館)
27||人博||741(吉田南総合図書館)
32192
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)教授 宇敷 重廣, 教授 森本 芳則, 准教授 木坂 正史
学位規則第4条第1項該当
Gossart, Luc. "Opérateurs de transfert de systèmes dynamiques partiellement hyperboliques aléatoires." Thesis, Université Grenoble Alpes, 2020. http://www.theses.fr/2020GRALM062.
In this thesis, we are interested in transfer operators associated with circle extensions of hyperbolic maps. We show a convergence in law of the flat traces of the reduced transfer operators, up to an Ehrenfest time, when the roof function is random
Moitier, Zoïs. "Étude mathématique et numérique des résonances dans une micro-cavité optique." Thesis, Rennes 1, 2019. http://www.theses.fr/2019REN1S053/document.
This thesis is devoted to the study of resonance frequencies of bidimensional optical cavities. More specifically, we are interested in whispering-gallery modes (modes localized along the cavity boundary with a large number of oscillations). The first part deals with the numerical computation of resonances by the finite element method using perfectly matched layers, and with a sensibility analysis in the three following situations: an unidimensional problem, a reduction of the rotationally invariant bidimensional case, and the general case. The second part focuses on the construction of asymptotic expansions of whispering-gallery modes as the number of oscillations along of boundary goes to infinity. We start by considering the case of a rotationally invariant problem for which the number of oscillations can be interpreted as a semiclassical parameter by means of an angular Fourier transform. Next, for the general case, the construction uses a phase-amplitude ansatz of WKB type which leads to a generalized Schrödinger operator. Finally, the numerically computed resonances obtained in the first part are compared to the asymptotic expansions made explicit by the use of a computer algebra software
Bader, Philipp Karl-Heinz. "Geometric Integrators for Schrödinger Equations." Doctoral thesis, Universitat Politècnica de València, 2014. http://hdl.handle.net/10251/38716.
Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716
TESIS
Premiado
Chabu, Victor. "Analyse semiclassique de l'équation de Schrödinger à potentiels singuliers." Thesis, Paris Est, 2016. http://www.theses.fr/2016PESC1029/document.
In the first part of this thesis we study the propagation of Wigner measures linked to solutions of the Schrödinger equation with potentials presenting conical singularities and show that they are transported by two different Hamiltonian flows, one over the bundle cotangent to the singular set and the other elsewhere in the phase space, up to a transference phenomenon between these two regimes that may arise whenever trajectories in the outsider flow lead in or out the bundle. We describe in detail either the flow and the mass concentration around and on the singular set and illustrate with examples some issues raised by the lack of uniqueness for the classical trajectories on the singularities despite the uniqueness of quantum solutions, dismissing any classical selection principle, but in some cases being able to fully solve the problem.In the second part we present a work in collaboration with Dr. Clotilde Fermanian and Dr. Fabricio Macià where we analyse a Schrödinger-like equation pertinent to the semiclassical study of the dynamics of an electron in a crystal with impurities, showing that in the limit where the characteristic lenght of the crystal's lattice can be considered sufficiently small with respect to the variation of the exterior potential modelling the impurities, then this equation is approximated by an effective mass equation, or, more generally, that its solution decomposes in terms of Bloch modes, each of them satisfying an effective mass equation specificly assigned to their Bloch energies
Miqueu, Jean-Philippe. "Étude des états fondamentaux du Laplacien magnétique en cas d'annulation locale du champ." Thesis, Rennes 1, 2016. http://www.theses.fr/2016REN1S061/document.
This thesis is devoted to the spectral analysis of the Schrödinger operator with magnetic field and semiclassical parameter, on a bounded regular domain in dimension two, with Neumann boundary condition. We investigate the case when the magnetic field vanishes along a union of smooth curves. The aim is to understand the influence of the cancellation and to study the behaviour of the lowest eigenvalues and the associated eigenfunctions when the semiclassical parameter tends to 0. In this regime - called the semiclassical limit - the precise description of the eigenpairs requires the understanding of underlying models. In the first part, we consider a magnetic field which vanishes linearly along a smooth simple curve intersecting the boundary. The second part is devoted to the case when the magnetic field vanishes quadratically. In both cases, we firstly give a one term asymptotics of the lowest eigenvalue. The upper bound is obtained by using appropriate test functions whereas the lower bound results from a localisation process. This last aspect constitutes the most difficult part because of the different scales involved. Then we investigate the localisation properties of the first eigenfunctions thanks to semiclassical Agmon estimates. This leads to a full asymptotic expansion of the first eigenvalues. In the case when the magnetic field vanishes quadratically, we study in addition the model operator for which the cancellation set is a union of two straight lines, whose intersection form a non-zero angle. In the small angle regime, the structure of the spectrum is governed by an operator symbol with two parameters. We establish different properties of this symbol and the associated band function. Numerical simulations based on the finite elements library Mélina++ have guided the analysis and illustrate the obtained results. The difficulties of the numerical computations - induced by the high phase oscillations of the eigenfunctions - are circumvented by polynomial interpolation of high degree
Arnoldi, Jean-François. "Résonances de Ruelle à la limite semiclassique." Thesis, Grenoble, 2012. http://www.theses.fr/2012GRENM105/document.
Since the work of Ruelle, then Rugh, Baladi, Tsujii, Liverani and others, it is kown that the convergence towards statistical equilibrium in many chaotic dynamical systems is gouverned by the Ruelle spectrum of resonances of the so-called transfer operator. Following recent works from Faure, Sjöstrand and Roy, this thesis gives a semiclassical approach for partially expanding chaotic dynamical systems. The first part of the thesis is devoted to compact Lie groups extenstions of expanding maps, essentially restricting to SU(2) extensions. Using Perlomov's coherent state theory for Lie groups, we apply the semiclassical theory of resonances of Helfer and Sjöstrand. We deduce Weyl type estimations and a spectral gap for the Ruelle resonances, showing that the convergence towards equilibrium is controled by a finite rank operator (as Tsujii already showed for partially expanding semi-flows). We then extend this approach to "open" models, for which the dynamics exhibits a fractal invariant reppeler. We show the existence of a discrete spectrum of resonances and we prove a fractal Weyl law, the classical analogue of Lin-Guillopé-Zworski's theorem on resonances of non-compact hyperbolic surfaces. We also show an asymptotic spectral gap. Finally we breifly explain why these models are interseting "toy models" to explore important questions of classical and quantum chaos. In particular, we have in mind the problem of proving lower bounds on the number of resonances, studied in the context of open quantum maps by Nonnenmacher and Zworski
Balança, Paul. "Régularité fine de processus stochastiques et analyse 2-microlocale." Phd thesis, Ecole Centrale Paris, 2014. http://tel.archives-ouvertes.fr/tel-00958290.
Deleporte-Dumont, Alix. "Low-energy spectrum of Toeplitz operators." Thesis, Strasbourg, 2019. http://www.theses.fr/2019STRAD004/document.
Berezin-Toeplitz operators allow to quantize functions, or symbols, on compact Kähler manifolds, and are defined using the Bergman (or Szeg\H{o}) kernel. We study the spectrum of Toeplitz operators in an asymptotic regime which corresponds to a semiclassical limit. This study is motivated by the atypic magnetic behaviour observed in certain crystals at low temperature. We study the concentration of eigenfunctions of Toeplitz operators in cases where subprincipal effects (of same order as the semiclassical parameter) discriminate between different classical configurations, an effect known in physics as quantum selection . We show a general criterion for quantum selection and we give detailed eigenfunction expansions in the Morse and Morse-Bott case, as well as in a degenerate case. We also develop a new framework in order to treat Bergman kernels and Toeplitz operators with real-analytic regularity. We prove that the Bergman kernel admits an expansion with exponentially small error on real-analytic manifolds. We also obtain exponential accuracy in compositions and spectra of operators with analytic symbols, as well as exponential decay of eigenfunctions
Karlsson, Ulf. "Semi-classical approximations of Quantum Mechanical problems." Doctoral thesis, KTH, Mathematics, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3440.
Keraval, Pierig. "Formules de Weyl par réduction de dimension : application à des Laplaciens électromagnétiques." Thesis, Rennes 1, 2018. http://www.theses.fr/2018REN1S093/document.
The thesis consists in the spectral study of partially semiclassical operators. When the geometry of the problem suggests an anisotropic localization of the eigenfunctions associated to low energies (boundary of the domain, vanishing magnetic field), the local expansion of the operator naturally brings to a doublescale structure. Via a reduction scheme "à la Born-Oppenheimer", using the formalism of pseudodifferential calculus for operator-valued symbols, we can show the existence of an effective operator, with scalar symbol. Then, we deduce Weyl formulae for the number of low-lying eigenvalues. This strategy is applied : to the Robin Laplacian on a bounded domain, in any dimension and to the magnetic Laplacian in R², in the case where the magnetic field vanishes on a closed curve
Alphonse, Paul. "Régularité des solutions et contrôlabilité d'équations d'évolution associées à des opérateursnon-autoadjoints." Thesis, Rennes 1, 2020. http://www.theses.fr/2020REN1S003.
The subject of this thesis deals with the sharp microlocal study of the smoothing and decreasing properties of evolution equations associated with two classes of non-selfadjoint operators with applications to the study of their subelliptic properties and to the null-controllability of these equations. The first class is composed of non-local operators given by the Ornstein-Uhlenbeck operators defined as the sum of a fractional diffusion and a linear transport operator. The second class is the class of accretive quadratic differential operators given by the Weyl quantization of complex-valued quadratic forms defined on the phase space with non-negative real parts. The aim of this work is to understand how the possible non-commutation phenomena between the self-adjoint and the skew-selfadjoint parts of these operators allow the associated semigroups to enjoy smoothing and decreasing properties in specific directions of the phase space that are explicitly described
Cekić, Mihajlo. "The Calderón problem for connections." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/267829.
Ourmières-Bonafos, Thomas. "Quelques asymptotiques spectrales pour le Laplacien de Dirichlet : triangles, cônes et couches coniques." Thesis, Rennes 1, 2014. http://www.theses.fr/2014REN1S143/document.
This thesis deals with the spectrum of the Dirichlet Laplacian in various two or three dimensional domains. First, we consider asymptotically flat triangles and cones with small aperture. These problems admit a semi-classical formulation and we provide asymptotic expansions at any order for the first eigenvalues and the associated eigenfunctions. These type of results is already known for thin domains with smooth profiles. For triangles and cones, we show that the problem admits now two different scales. Second, we study a family of conical layers parametrized by their aperture. Again, we consider the semi-classical limit when the aperture tends to zero: We provide a two-term asymptotics of the first eigenvalues and we prove a localization result about the associated eigenfunctions. We also estimate, for each chosen aperture, the number of eigenvalues below the threshold of the essential spectrum