Functions of linear operators: Parameter differentiation

We derive a useful expression for the matrix elements [partial derivative f [A(t)]/partial derivative t](ij) of the derivative of a function f[A(t)] of a diagonalizable linear operator A(t) with respect to the parameter t at t (0). The function f[A(t)] is supposed to be an operator acting on the same space as the operator A(t) which is assumed to have a nondegenerate, pure p oint spectrum. We use the basis which diagonalizes A(t(0)), i.e., [A(t(0)]( ij)=lambda(i)delta(ij), and obtain [partial derivative f[A(t)]/partial deri vative t parallel to(t=t0)](ij)=[partial derivative A/partial derivative t parallel to(t=t0)](ij){[f(lambda(j))-f(lambda(i))]/(lambda(j)-lambda(i))}. In addition to this, we show that further elaboration on the (not necessari ly simple) integral expressions given by Wilcox (who basically considered f [A(t)] of the exponential type) and generalized by Rajagopal [who extended Wilcox results by considering f[A(t)] of the q-exponential type where exp(q )(x)=[1+(1-q)x](1/(1-q)) with q is an element of R; hence, exp(1)(x)=exp(x) ] yields these same expressions. Some of the lemmas first established by th e above authors are easily recovered. (C) 2000 American Institute of Physic s. [S0022-2488(00)02205-2].