Matrix Finite-Zone Dirac-Type Equations

Abstract

Weyl–Titchmarsh matrix functions play an essential role in the spectral theory of Dirac-type equations ( Oper. Theory: Adv. Appl. 107 (1999)). In this paper, we have constructed a class of Weyl–Titchmarsh matrix-functions generating potentials of finite-zone type. It has turned out that the corresponding potentials have derivatives of an arbitrary order. Using the above-mentioned results, we deduce the matrix analogue of the trace formula for finite-zone matrix potentials. In the last part of the paper, we consider separately the scalar case of Dirac-type equations. For this case, we have constructed finite-zone potentials in explicit forms and proved that these potentials are quasiperiodical. We note that for scalar Schrödinger equations the corresponding results are well known (see Invent. Math. 30 (1975), 217–274; “Soliton and the Inverse Scattering Transform,” SIAM, Philadelphia, 1981; Rev. Sci. Technol. 23 (1983), 20–50; “Theory of Solitons, The Method of Inverse Problem,” New York, 1984; “Inverse Sturm–Liouville Problems,” VSP, Zeist, 1987; “Inverse Spectral Theory,” Academic Press, New York, 1987).