# Eikonal Blog

## 2010.01.22

### BCS formula on computer

Filed under: mathematics — Tags: , , — sandokan65 @ 16:06

Def: $e^C :\equiv e^A e^B$

$C = A + \int_0^1 dt \psi(e^{a}e^{t b}) B$,
where $a:\equiv ad(A)$, $b:\equiv ad(B)$ and
$\psi(z):\equiv \frac{z \ln(z)}{z-1} = 1 + \frac{w}{1\cdot 2} - \frac{w^2}{2\cdot 3} + \frac{w^3}{3\cdot 4} - \cdots$ for $w:\equiv z-1$.

Richtmyer and Greenspan obtained the 512 initial terms of that expansion, but here I present only several of them:

$C = A + B + \frac12 a B + \frac1{12} a^2 B - \frac1{12} b a B + \frac1{12,096} a^4 b^2 a B - \frac1{6,048} a^3 b a^3 B - \frac1{3,780} a^3 b a b a B + ...$

Not all terms are independent, due to Jacobi identities.

Sources:

• T1267: R.D. Richtmyer and S. Greenspan: “Expansion of the Campbell-Baker-Hausdorff formula by Computer”; Communications on Pure and Applied Mathematics, Vol XVIII 107-108 (1965).

### Guenin on commutators

Filed under: Uncategorized — Tags: , — sandokan65 @ 15:46

$A B^n = \sum_{j=0}^n \binom{n}{j} (-)^j B^{n-j} \ ad(B)^j A.$

$B^n A = \sum_{j=0}^n \binom{n}{j} (-)^j \ (ad(B)^j A) B^{n-j}.$

$\partial_s f(F(s)) = \sum_{j=0}^\infty \frac{(-)^j}{(j+1)!} f^{(j+1)}(F(s)) \ ad(F(s)^j \partial_s F(s) = \sum_{j=0}^\infty \frac{1}{(j+1)!} \ (ad(F(s))^j \partial_s F(s) ) f^{(j+1)}(F(s)).$

$[H, f(A)] = \sum_{j=0}^\infty \frac{(-)^j}{j!} f^{(j)}(A) \ ad(A)^j H = - \sum_{j=0}^\infty \frac1{j!}\ (ad(A)^j H) f^{(j)}(A).$
—-
Sources:

• T1266: M. Guenin: “On the Derivation and Commutation of Operator Functionals”; Helv. Phys. Act. 41, 75-76, 1968.