A Kahlerian approche to the Schrodinger equation in Siegel jacobi Space of the lognormal distribution

Abstract

In this paper, we describe the evolution of spectral curves in the Siegel Jacobi space through the Schrodinger equation constructed from a Kahler geometry induced on the lognormal statistical manifold via Dombrowski's construction. We introduce new holomorphic structures and show that the Hamiltonian vector field coincides with the fundamental vector field generated by holomorphic isometries. We construct the time dependent Schrodinger equation from this geometric setting and show that the associated energy is not constant, but varies with time. This work establishes a bridge between Kahler geometry, statistical models, and the formalism of quantum mechanics.