Journal of Mathematical Physics

Analytic representations with theta functions for systems on ℤ(d) and on 𝕊

Pavlos Evangelides, Chao Lei, Apostol Vourdas

Published Jul 1, 2015• 3 citations

Abstract

An analytic representation with theta functions on a torus, for systems with variables in ℤ(d), is considered. Another analytic representation with theta functions on a strip, for systems with positions in a circle 𝕊 and momenta in ℤ, is also considered. The reproducing kernel formalism for these two systems is studied. Wigner and Weyl functions in this language are also studied.

DOI: 10.1063/1.4927256

Related papers

Analytic representations with Theta functions for systems on Z(d) and on S

Pavlos Evangelides, Chao Lei, Apostol Vourdas

An analytic representation with Theta functions on a torus, for systems with variables in Z(d), is considered. Another analytic representation with Theta functions on a strip, for systems with positions in a circle S and momenta in Z, is also considered. The reproducing kernel formalism for these two systems is studied. Wigner and Weyl functions in this language, are also studied

arXiv:1502.05309Feb 18, 20150 citations

Analytic representations of quantum systems with Theta functions

Pavlos Evangelides

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

Jan 1, 20150 citations

Zeros in analytic representations of finite quantum systems on a torus

M. Tubani, Apostol Vourdas, S Zhang

An analytic representation of finite quantum systems with positions and momenta in is considered. Quantum states are represented by analytic functions on a torus, which have exactly d zeros. The d paths ζ n ( t ) of the zeros on the torus, as the system evolves in time, define the quantum system. Special cases where of the zeros follow the same path are also studied. The general ideas are demonstrated with several examples.

Physica ScriptaAug 18, 20106 citations

Analytic Representations of Finite Quantum Systems on a Torus

Muna Jabuni

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

Jan 1, 20100 citations

Open Problems On Zeros Of Analytic Functions In Finite Quantum Systems

Muna Tabuni

The paper contains an investigation on basic problems about the zeros of analytic theta functions. A brief introduction to analytic representation of finite quantum systems is given. The zeros of this function and there evolution time are discussed. Two open problems are introduced. The first problem discusses the cases when the zeros follow the same path. As the basis change the quantum state |f transforms into different quantum state. The second problem is to define a map between two toruses where the domain and the range of this map are the analytic functions on toruses.

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer EngineeringAug 22, 20131 citation

Analytic representation of quantum systems

Hend Abdelgader Eissa

Finite quantum systems with d-dimension Hilbert space, where position x and \nmomentum p take values in Zd(the integers modulo d) are studied. An analytic \nrepresentation of finite quantum systems, using Theta function is considered. \nThe analytic function has exactly d zeros. The d paths of these zeros on the \ntorus describe the time evolution of the systems. The calculation of these \npaths of zeros, is studied. The concepts of path multiplicity, and path winding \nnumber, are introduced. Special cases where two paths join together, are also \nconsidered. A periodic system which has the displacement operator to real \npower t, as time evolution is also studied. \nThe Bargmann analytic representation for infinite dimension systems, with \nvariables in R, is also studied. Mittag-Leffler function are used as examples of \nBargmann function with arbitrary order of growth. The zeros of polynomial \napproximations of the Mittag-Leffler function are studied.

Jan 1, 20161 citation