Abstract

This paper aims to prove the Riesz basisness of root functions of the non-selfadjoint a discontinuous Sturm–Liouville operator with periodic boundary condition which are not strong regular and with conjugate conditions. It is assumed that the potentials of differential operator are complex valued and continuously differentiable functions and both conjugate conditions have different finite one-sided limits at point zero. In order to prove Riesz basisness of root functions, we firstly acquire asymptotic expressions of fundamental solutions. By using these solutions in the characteristic determinant, it is obtained asymptotic formulas of eigenvalues by means of Rouche theorem. Then by the aid of asymptotic formulas of eigenfunctions,Riesz basisness is shown. it is also proved the Riesz basisness of root functions of the same operator with antiperiodic boundary conditions and with same conjugate conditions.

中文翻译：

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带共轭条件的Sturm-Liouville算子根函数的Riesz基性

摘要

本文旨在证明具有周期边界条件的非自伴非连续Sturm-Liouville算子的根函数的Riesz基性，该周期边界条件不是强正则且具有共轭条件。假定微分算子的电位是复数值，并且是连续可微的函数，并且两个共轭条件在零点处都有不同的有限一侧限制。为了证明根函数的Riesz基性，我们首先获得基本解的渐近表达式。通过在特征行列式中使用这些解，可以借助Rouche定理获得特征值的渐近公式。然后借助特征函数的渐近公式，证明了Riesz基性。