Analytic Representations of Finite Quantum Systems on a Torus

Pavlos Evangelides

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.

Hend Abdelgader Eissa, Pavlos Evangelides, Chao Lei +1 more

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.

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.

Vladimir V. Kornyak