Quantum systems with a finite Hilbert space, where position x and momen- tum p take values in Z(d) (integers modulo d), are studied. An analytic representation of finite quantum systems is considered. Quantum states are represented by analytic functions on a torus. This function has exactly d zeros, which define uniquely the quantum state. The analytic function of a state can be constructed using its zeros. As the system evolves in time, the d zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, for various Hamiltonians, are given. In addition, for given paths ³n(t) of the d zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum systems are considered. Special cases where M of the zeros follow the same path are also studied, and general ideas are demonstrated with several ex- amples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It is evidenced within the study that a small perturbation of the initial values of the zeros splits a path with multiplicity M into M different paths. / Libyan Cultural Affairs

No description available.

Quantum systems with a finite Hilbert space, where position x and momen- tum p take values in Z(d) (integers modulo d), are studied. An analytic representation of finite quantum systems is considered. Quantum states are represented by analytic functions on a torus. This function has exactly d zeros, which define uniquely the quantum state. The analytic function of a state can be constructed using its zeros. As the system evolves in time, the d zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, for various Hamiltonians, are given. In addition, for given paths ³n(t) of the d zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum systems are considered. Special cases where M of the zeros follow the same path are also studied, and general ideas are demonstrated with several ex- amples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It is evidenced within the study that a small perturbation of the initial values of the zeros splits a path with multiplicity M into M different paths.

No description available.

Thèse de doctorat : Sciences mathématiques : Paris, Faculté des sciences : 1858. / Titre provenant de l'écran-titre.

Thèse--Paris.

In the present work we develop a unified way of looking at normed spaces of analytic functions (NSAF's) and their embedding into the Frechet space of analytic functions on a general domain, by requiring only that the embedding map is bounded. This is a succinct definition of NSAF and derive from it a list of interesting properties. For example Proposition 4.4 describes the behavior of point evaluations and Proposition 4.6 part (i) gives a general sufficient condition for a NSAF to be a Banach space, which as far as we know, are new results. Also, Proposition 4.5, parts (ii) and (iii) of Proposition 4.6 and Proposition 4.7 are results, which are slight generalizations of fairly standard results, which show up elsewhere in a more specific setting. Some of the facts about NSAF's are stated and proven in a more general context. In particular, a significant part of the material is dedicated to the normed space of continuous functions on a metric space. On the other hand, we provide the necessary background on differential geometry and complex analysis, which further determine the peculiarities in the context of spaces of analytic functions. At the end we illustrate our results on two specific examples of NSAF's, namely the Bergman and the Bloch Spaces over a general domain in Cd. We give a new proof of the reflexivity of the Bergman Space Ap(G, μ) for the case p>1 and of the Schur property of A1(G, μ). We also give new proofs for the equivalences of some of the definitions of the Bloch functions.

In the present work we develop a unified way of looking at normed spaces of analytic functions (NSAF's) and their embedding into the Frechet space of analytic functions on a general domain, by requiring only that the embedding map is bounded. This is a succinct definition of NSAF and derive from it a list of interesting properties. For example Proposition 4.4 describes the behavior of point evaluations and Proposition 4.6 part (i) gives a general sufficient condition for a NSAF to be a Banach space, which as far as we know, are new results. Also, Proposition 4.5, parts (ii) and (iii) of Proposition 4.6 and Proposition 4.7 are results, which are slight generalizations of fairly standard results, which show up elsewhere in a more specific setting. Some of the facts about NSAF's are stated and proven in a more general context. In particular, a significant part of the material is dedicated to the normed space of continuous functions on a metric space. On the other hand, we provide the necessary background on differential geometry and complex analysis, which further determine the peculiarities in the context of spaces of analytic functions. At the end we illustrate our results on two specific examples of NSAF's, namely the Bergman and the Bloch Spaces over a general domain in Cd. We give a new proof of the reflexivity of the Bergman Space Ap(G, μ) for the case p>1 and of the Schur property of A1(G, μ). We also give new proofs for the equivalences of some of the definitions of the Bloch functions.

Quantum systems in a d-dimensional Hilbert space are considered, where the phase spase is Z(d) x Z(d). An analytic representation in a cell S in the complex plane using Theta functions, is defined. The analytic functions have exactly d zeros in a cell S. The reproducing kernel plays a central role in this formalism. Wigner and Weyl functions are also studied. Quantum systems with positions in a circle S and momenta in Z are also studied. An analytic representation in a strip A in the complex plane is also defined. Coherent states on a circle are studied. The reproducing kernel is given. Wigner and Weyl functions are considered.

In this thesis, we consider linear extremal problems in the Hp spaces. For many of these extremal problems, a unique solution can be guaranteed. We will examine some of the classical examples of extremal problems in these spaces. With this framework in place we will then consider a particular problem which does not always have a unique solution.