Factorization on a Riemann Surface in Scattering Theory

Motivated by a problem of scattering theory, the authors solve by quadratures a vector Riemann– Hilbert problem with the matrix coefficient of Chebotarev–Khrapkov type. The problem of matrix factorization reduces to a scalar Riemann–Hilbert boundary-value problem on a twosheeted Riemann surface of genus 3 that is topologically equivalent to a sphere with three handles. The conditions quenching an essential singularity of the solution at infinity lead to the classical Jacobi inversion problem. It is shown that this problem is equivalent to an algebraic equation of degree that coincides with the genus of the Riemann surface. A closedform solution of this nonlinear problem is found for genus 3. A normal matrix of factorization and the canonical matrix are constructed in explicit form. It is proved that the vector Riemann– Hilbert problem possesses zero partial indices and is, therefore, stable. The proposed technique is illustrated by a problem of scattering of sound waves by a perforated sandwich panel.