Analytic functions of the annihilation operator

TL;DR

This paper introduces a novel method for constructing analytic functions of the annihilation operator, utilizing Runge's theorem and dyad representations, expanding the mathematical tools available in quantum operator analysis.

Contribution

It presents the first method for defining analytic functions of the annihilation operator on specific domains using Runge's polynomial approximation.

Findings

Provides a new form of the identity suited for the domain of the function

Defines functions of the annihilation operator on compact domains that do not separate the complex plane

Expresses the constructed operators in terms of dyads of Fock states

Abstract

A method for construction of analytic function f of the annihilation operator is given for the first time. f(z) is analytic on some compact domain that does not separate the complex plane. A new form of the identity is given, which is well suited for domain of function. Using Runge's polynomial approximation theorem, such a function f of the annihilation operator is defined on the whole domain. The constructed operators are given in terms of dyads formed of Fock states.