Regular spectral problems for systems of ordinary differential equations of the first order

Abstract

Here y(x) = (y1(x), . . . , yn(x)) and A(x) = {ajk(x)}j,k=1, where the ajk are complex functions in L1[0, 1], B = diag{b1, . . . , bn} with 0 ̸= bj ∈ C, ρ is a uniformly positive bounded measurable function, U0 and U1 are n × n number matrices, and λ is the spectral parameter. Special cases of such problems include spectral problems for the Dirac operator (which corresponds to n = 2, b1 = −b2 = i, and ρ(x) ≡ 1) and for the systems of telegrapher’s equations (n = 2, b1 = −b2 = 1, and ρ ∈ L1[0, 1]). We consider the case when all the bj are different. The straight lines Re(bkλ) = Re(bjλ), k ̸= j = 1, . . . , n, partition the complex plane C into (at most n − n) sectors Γl with vertex at the origin. In each Γl, 1 ⩽ l ⩽ n − n, we have a fundamental matrix of solutions of (1), which has the form Y(x, λ) = E(x, λ)(M(x) + R(x, λ)), where