Representations of the Schrodinger algebra and Appell systems

TL;DR

This paper explores the structure and representations of the Schrödinger algebra using Appell systems, constructing a Hilbert space with probabilistic interpretations and analyzing related evolution equations.

Contribution

It introduces a novel representation of the Schrödinger algebra via Appell systems and constructs associated Hilbert spaces with probabilistic and orthogonal basis structures.

Findings

Identified a Leibniz function for the algebra

Constructed orthogonal basis for the Hilbert space

Derived Appell systems related to evolution equations

Abstract

We investigate the structure of the Schrodinger algebra and its representations in a Fock space realized in terms of canonical Appell systems. Generalized coherent states are used in the construction of a Hilbert space of functions on which certain commuting elements act as self-adjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting feature is how the structure of the Lie algebra is reflected in the probability density function. A Leibniz function and orthogonal basis for the Hilbert space is found. Then Appell systems connected with certain evolution equations, analogs of the classical heat equation, on this algebra are computed.