Electronic Resource
College Park, Md.
:
American Institute of Physics (AIP)
Journal of Mathematical Physics
41 (2000), S. 3278-3282
ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
We derive a useful expression for the matrix elements [∂f[A(t)]/∂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 t0. 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 point spectrum. We use the basis which diagonalizes A(t0), i.e., [A(t0]ij=λiδij, and obtain [∂f[A(t)]/∂t|t=t0]ij=[∂A/∂t|t=t0]ij{[f(λj)−f(λi)]/(λj−λi)}. In addition to this, we show that further elaboration on the (not necessarily 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 expq(x)≡[1+(1−q)x]1/(1−q) with q∈R; hence, exp1(x)=exp(x)] yields these same expressions. Some of the lemmas first established by the above authors are easily recovered. © 2000 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.533305
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