Functions of linear operators: Parameter differentiation

TL;DR

This paper derives a formula for the parameter derivative of a function of a diagonalizable linear operator, generalizing previous integral expressions for exponential and q-exponential functions.

Contribution

It provides a new explicit expression for the matrix elements of the derivative of operator functions, extending prior integral formulas to more general functions.

Findings

Derived a formula for the derivative of operator functions in a diagonal basis.

Generalized Wilcox's integral expressions to q-exponential functions.

Recovered lemmas from Wilcox 1967 and Rajagopal 1998 as special cases.

Abstract

We derive a useful expression for the matrix elements $[\frac{\partial f[A(t)]}{\partial t}]_{i j}$ of the derivative of a function $f[A(t)]$ of a diagonalizable linear operator $A(t)$ with respect to the parameter $t$. The function $f[A(t)]$ is supposed to be an operator acting on the same space as the operator $A(t)$. We use the basis which diagonalizes A(t), i.e., $A_{i j}=\lambda_i \delta_{i j}$, and obtain $[\frac{\partial f[A(t)]}{\partial t}]_{i j}=[\frac{\partial A}{\partial t}]_ {i j}\frac{f(\lambda_j) - f(\lambda_i)} {\lambda_j - \lambda_i}$. In addition to this, we show that further elaboration on the (not necessarily simple) integral expressions given by Wilcox 1967 (who basically considered $f[A(t)]$ of the exponential type) and generalized by Rajagopal 1998 (who extended Wilcox results by considering $f[A(t)]$ of the $q$-exponential type where $\exp_q(x) \equiv [1+(1-q)x]^{1/(1-q)}$ with $q \in {\cal {R}}$; hence, $\exp_1 (x)=\exp(x))$ yields this same expression. Some of the lemmas first established by the above authors are easily recovered.