题目： Galoisian approach to complex oscillation theory of some Hill equations

报告人：虞国富 教授 

（上海交通大学）

Abstract : We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponential of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. That are solutions obtainable by suitable differential field extensions construction. In particular, we have established a full correspondence between solutions of non-oscillatory type  and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential is a combination of four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions in C that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lame and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago. This is a joint a work with Yik-Man Chiang.

时间： 12月28日（周五）11:00-12:00

地点：565net必赢客户端本部教二楼 627 教室

联系人：李春霞

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