Analytic representation of quantum systems

Abstract

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