Abstract :
In this note, we prove that the maximally defined operator associated with the Dirac-type differential
expression
M(Q) = i d
dx Im −Q
−Q∗ − d
dx Im ,
where Q represents a symmetric m × m matrix (i.e., Q(x) = Q(x) a.e.) with entries in L1
loc(R),
is J-self-adjoint, where J is the antilinear conjugation defined by J = σ1C, σ1 = 0 Im
Im 0 and
C(a1, . . ., am,b1, . . ., bm) = (a1, . . . , am, b1, . . . , bm) . The differential expression M(Q) is of
significance as it appears in the Lax formulation of the non-abelian (matrix-valued) focusing nonlinear
Schrödinger hierarchy of evolution equations.
2004 Elsevier Inc. All rights reserved.
