J -self-adjointness of a Class of Dirac-type Operators

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 L 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 nonabelian (matrix-valued) focusing nonlinear Schrödinger hierarchy of evolution equations. To set the stage for this note, we briefly mention the Lax pair and zero-curvature representations of the matrix-valued Ablowitz-Kaup-Newell-Segur (AKNS) equations and the special focusing and defocusing nonlinear Schrödinger (NLS) equations associated with it. Let P = P (x, t) and Q = Q(x, t) be smooth m × m matrices, m ∈ N, and introduce the Lax pair of 2m× 2m matrix-valued differential expressions M(P,Q) = i ( d dx Im −Q P − d dx Im ) (1) L(P,Q) = i ( d dx Im − 1 2QP −Q d dx − 1 2Qx P d dx − 1 2Px − d dx Im + 1 2PQ ) (2) and the 2m× 2m zero-curvature matrices U(z, P,Q) = ( −izIm Q P izIm ) , (3) V (z, P,Q) = ( −izIm − i 2QP zQ+ i 2Qx zP − i 2Px iz Im + i 2PQ ) , (4) where z ∈ C denotes a (spectral) parameter and Im is the identity matrix in C . Then the Lax equation d dt M − [L,M ] = 0 (5) is equivalent to the m×m matrix-valued AKNS system Qt − i 2 Qxx + iQPQ = 0, Pt + i 2 Pxx − iPQP = 0, (6) 1991 Mathematics Subject Classification. Primary: 34L40. Secondary: 35Q55.