TD 12 : Algèbre des opérateurs.

A, B, C
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
O f(O)
f(x)x= 0
g(O)f(O)A B f(A)
g(B)
Lx=Y ∂z−Z∂yLy
Lz(x, y, z)
[Lx, Ly] = −Lz
L2=L2
x+L2 y+L2 zL2Lµµ=x, y, z
d
dt exp(tA) = Aexp(tA)
A
exp(P−1AP ) =
P−1exp(A)P A, P
A=0−x
x0
A2Ann
exp(A) = cos x−sin x
sin xcos x

sin cos
C=0−x
0 0 , D =0 0 x0
C2=D2= 0 eC.eDeCeD
eC+D
A B
e(A+B)t=eAt +Zt
0
eA(t−s)Be(A+B)sds
y0−ay =f(x)
y=eaty0+Zt
0
ea(t−s)f(s)ds
U(t) = exp((A+B)t)
U
exp(At)B
∂2u
∂t2−c2∂2u ∂x2= 0
u(x, 0) = f(x)∂tu(x, t)|t=0 =g(x)
∂2u/∂t2−c2D2u= 0
u00 −a2u= 0
D In
In(x) = 1
πZπ
0
excos θcos(nθ)dθ
I0
n(x) = (1/2) (In−1(x) + In+1(x))
exp(−x)In(x)n= 0,1, ...
D