Student Seminar: Classical and Quantum Integrable Systems

Thus,
g(λ)Q(λ)g(λ) −1 =
(
= g 0 + (λ − λ k )g 1 + · · ·
(
= g 0 Q 0 g0 −1 + (λ − λ k )
Thus, we see that g 0 must diagonalize Q 0 :
and g 1 is found from the condition that
)(
)(
Q 0 + (λ − λ k )Q 1 + · · · g −1
g 0 Q 1 g0 −1 + g 1 Q 0 g0 −1 − g 0 Q 0 g0 −1 g 1 g0
−1
D 0 = g 0 Q 0 g −1
0
)
0 − (λ − λ k )g0 −1 g 1 g0 −1 + · · ·
)
+ · · ·
g 0 Q 1 g −1
0 + g 1 Q 0 g −1
0 − g 0 Q 0 g −1
0 g 1 g −1
0 = g 0 Q 1 g −1
0 + [g 1 g −1
0 , D 0 ]
is diagonal. The commutator of a diagonal matrix with any matrix is off-diagonal.
Thus, [g 1 g0 −1 , D 0 ] is off-diagonal and the matrix g 1 is found from the condition that
[g 1 g0 −1 , D 0 ] kills the off-diagonal elements of g 0 Q 1 g0 −1 . Thus,
D(λ) = D 0 + (λ − λ k ) ( )
g 0 Q 1 g0
−1 E ii ii + · · ·
Thus, we have shown that by means of a regular similarity transformation around
the pole λ = λ k the Lax matrix can be brought to the diagonal form
A(λ) =
∑−1
A
}{{} k,r
r=−n k diag
(λ − λ k ) r + regular
The diagonalizing matrix g(λ) is defined up to right multiplication by an arbitrary
analytic diagonal matrix.
Define the matrix B(λ) as
M(λ) = g(λ)B(λ)g(λ) −1 +
˙ g(λ)g(λ) −1 ,
where g(λ) is a regular matrix which diagonalizes L(λ) around λ = λ k . The Lax
representation implies that
Ȧ(λ) = [B(λ), A(λ)] .
Since A(λ) is diagonal then A(λ) ˙ = 0, i.e., A(λ) comprises integrals of motion!
Further, the consistency of the Lax equation implies that B(λ) is a diagonal matrix
as well.
We have
L k = (g (k) A (k) g (k)−1 ) − , M k = (g (k) B (k) g (k)−1 ) − .
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