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

TL;DR

This paper explores the geometric formulation of the Schrödinger equation within the Siegel Jacobi space of the lognormal distribution, linking Kahler geometry, statistical models, and quantum mechanics.

Contribution

It introduces a novel Kahler geometric approach to the Schrödinger equation on the lognormal manifold, including new holomorphic structures and dynamic energy analysis.

Findings

Spectral curves evolve in Siegel Jacobi space via a Kahler geometric Schrödinger equation.

The Hamiltonian vector field aligns with holomorphic isometries.

Energy varies with time in this geometric quantum framework.

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.