Sistemas quânticos cuja dinâmica é não-unitária e que evoluem adiabaticamente apresentam características únicas com aplicações no campo da computação quântica. Estudamos nessa dissertação o formalismo de sistemas quânticos abertos, a teoria de semigrupos dinâmicos e os chamados operadores de Lindblad. Enunciamos e provamos o teorema adiabático na formulação de T. Kato a fim de entender a idéia e o formalismo por trás de regimes adiabáticos. Utilizamos essas ferramentas para descrever o problema de otimização de trajetórias adiabáticas em sistemas quânticos dissipativos (cuja dinâmica é dada por uma classe de operadores de Lindblad) e, seguindo as indicações de Avron et al. [8], obtemos as condições para que essa otimização seja única e aplicamos esse resultado em algoritmos quânticos de busca. / Quantum systems whose dynamics is non-unitary and develop adiabatically exhibit unique characteristics with applications in the field of quantum computing. We study in this dissertation formalism of open quantum systems, the theory of dynamical semigroups and called Lindblad operators. We state and prove the adiabatic theorem in Kato T. formulation in order to understand the idea and the formalism behind adiabatic regimes. We use these tools to describe the adiabatic trajectory optimization problem in dissipative quantum systems (whose dynamics is given by a Lindblad operator class) and following the advice of Avron et al. [8], we obtain the conditions for this optimization is unique and apply this result in search of quantum algorithms.

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We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

<div>We study the Gaudin model associated to Lie algebras of classical types.</div><div><br></div><div>First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.</div><div><br></div><div>Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian. </div>

Mirror symmetry is the phenomenon, originally discovered by physicists, that Calabi-Yau manifolds come in dual pairs, with each member of the pair producing the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. One important problem in mirror symmetry is that there may be several ways to construct a mirror dual for a Calabi-Yau manifold. Hence it is a natural question to ask: when two different mirror symmetry constructions apply, do they agree?We specifically consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry. BHK mirror symmetry was inspired by the LandauGinzburg/Calabi-Yau correspondence, while LPK3 mirror symmetry is more classical. In particular, for algebraic K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani Boissi`ere-Sarti originally showed that they agree when n = 2, and more recently Comparin-Lyon-Priddis-Suggs showed that they agree when n is prime. However, the n being composite case required more sophisticated methods. Whenever n is not divisible by four (or n = 16), this problem was solved by Comparin and Priddis by studying the associated lattice theory more carefully. In this thesis, we complete the remaining case of the problem when n is divisible by four by finding new isomorphisms and deformations of the K3 surfaces in question, develop new computational methods, and use these results to complete the investigation, thereby showing that the BHK and LPK3 mirror symmetry constructions also agree when n is composite.

Recently Wulfman found great merit in Barut's idea on atomic super-multiplets, and he introduced the concept of the generalized Hamiltonian that is the Hamiltonian of all atoms. 24 Investigating the Schrodinger equation with this generalized Hamiltonian, it should be possible to relate the properties of different atoms and find the structure of the periodic chart from fundamental principles of dynamics and group theory. One can can use the same kinds of methods for relating the properties of different states of a single hydrogen atom with the aid of the degeneracy group S0 (4) and dynamical group SO (4, 2). These groups represent the symmetries of the time-independent and time-dependent Schrodinger equations with ordinary Hamiltonian.(25,26) The idea then is to apply these methods to the system defined by a generalized Hamiltonian. In chapter II of this thesis, we will consider the classification of chemical elements, in the light of the concept of the generalized Hamiltonian. We will make a group theoretical classification based on the characteristics of the outermost electrons in the central-field model of atomic ground states. We conclude that the classification group may be SO (p,q) with p+q≧, p ≧4. In chapter III of this thesis, we will review Wulfman's work briefly and consider an application of his idea to the ground state of helium making use of the group SO (4,1) xSO (4,1). We arrive at the conclusion that we can obtain physically significant configuration mixing using SO (4,1) xSO (4,1) or SO (4,2) xSO (4,2) in a manner analogous to the way in which SO (4) xSO (4) is used to determine configuration mixing in doubly excited states of helium-like systems.

A group theoretic approach to Layzer's 1/2 expansion method is explored. In part this builds on earlier work of Wulfman(2), of Moshinsky et al(l4), and of Sinanoglu, Herrick(lS), and Kellman (16) on second row atoms. I investigate atoms with electrons in the 3s-3p-3d shell and find: 1. Wulfman's constant of motion accurately predicts configuration mixing for systems with two to eight electrons in the 3s-3p subshell. 2. The same constant of motion accurately predicts configuration mixing for systems with two electrons in the 3s-3p-3d shell. 3. It accurately predicts configuration mixing in systems of high angular momentum L and of high spin angular momentum S containing three electrons in the 3s-3p-3d shell, but gives less accurate results when L and S are both small. I also show how effective nuclear charges may be calculated by a group theoretical approach. In addition I explore several new methods for expressing electron repulsion operators in terms of operators of the 80(4,2) dynamical group of one - electron atoms.

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第17837号 / 人博第658号 / 新制||人||158(附属図書館) / 25||人博||658(吉田南総合図書館) / 30652 / 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 上木 直昌, 教授 森本 芳則, 教授 髙﨑 金久 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

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A dynamical system originally invented by Boltzmann has had recent developments. The system consists of a particle in a gravitational potential with an added centrifugal force, which is subject to reflection against a wall that separates the system from the gravitational center. The recent developments are with regards to the integrability of the system in the special case of vanishing centrifugal term. The purpose of this essay is to explicate these developments. / Ett dynamiskt system, ursprungligen uppfunnet av Boltzmann, har nyligen sett utvecklingar. Systemet består av en partikel i en gravitationspotential med en tillagd centrifugalkraft, som reflekterar vid kontakt med en vägg som skiljer partikeln och gravitationscentrumet. De nya utvecklingarna är inom systemets integrabilitet i det specialfall att centrifugalkraften är borttagen. Syftet med denna uppsats är att explicera dessa framtaganden.