Spectrum of the Riemann–Hilbert–Poincaré problem for analytic functions

Abstract

We study the Riemann–Hilbert–Poincaré boundary value problem for analytic function. This problem will lead to inhomogeneous Fuchsian differential equations. We find that its spectrum is not characterized by the smoothness of its coefficient on the boundary but by its interior analytic continuation property. Moreover, the multiplicities of eigenfunctions for different eigenvalues are not necessarily the same even when the eigenvalues are small.

Keywords:

• Boundary value problem
• Analytic function
• Inhomogeneous Fuchsian differential equations
• AMS 2000 Mathematics Subject Classification: 30E25

Acknowledgment

This research was partly supported by the National Natural Science Foundation of China and Natural Science Foundation of Guangdong.

Notes

Dedicated to Heinrich Begehr on the occasion of his 65th birthday.

Additional information

Notes on contributors

Ming-Sheng Liu

Dedicated to Heinrich Begehr on the occasion of his 65th birthday.