Letters in Mathematical Physics

, Volume 104, Issue 11, pp 1365–1384 | Cite as

The Baker–Akhiezer Function and Factorization of the Chebotarev–Khrapkov Matrix

  • Yuri A. AntipovEmail author


A new technique is proposed for the solution of the Riemann–Hilbert problem with the Chebotarev–Khrapkov matrix coefficient \({G(t) = \alpha_{1}(t)I + \alpha_{2}(t)Q(t)}\), \({\alpha_{1}(t), \alpha_{2}(t) \in H(L)}\), I = diag{1, 1}, Q(t) is a \({2\times2}\) zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of essential singularities of the solution to the associated homogeneous scalar Riemann–Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker–Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann–Hilbert problem requires the finding of the \({\rho}\) zeros of the Baker–Akhiezer function (\({\rho}\) is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-\({\rho}\) polynomial and solution of a certain linear algebraic system of \({\rho}\) equations.

Mathematics Subject Classification

30E25 30F99 45E 


Riemann–Hilbert problem Baker–Akhiezer function Riemann surfaces 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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