Trace Formulae for Matrix Integro-Differential Operators

where λ is a spectral parameter, Y (x) = [yk(x)]k=1,d is a column vector, Q(x) and M(x, t) are d×d real symmetric matrix-valued functions, and h and H are d×d real symmetric constant matrices. M(x, t) is an integrable function on the set D0 def ={(x, t) : 0≤ t ≤ x ≤ π, x, t ∈ R}, Q ∈ C1[0,π], where C1[0,π] denotes a set whose element is a continuously differentiable function on [0,π]. In particular, h = ∞ in (2) means the Dirichlet boundary condition Y (0) = 0, and H = ∞ in (3) means the Dirichlet boundary condition Y (π) = 0. For the matrix Sturm–Liouville equation (when M = 0 in (1)) properties of spectral characteristics were provided in [1 – 4], and asymptotics of eigenvalues for the integro-differential operator with d = 1 in (1) were given in [5 – 9]. Gelfand and Levitan [10] discussed the Sturm– Liouville problem