On the estimations of the small eigenvalues of Sturm\u2013Liouville operators with periodic and antiperiodic boundary conditions

Cemile Nur

doi:10.1186/s13661-018-1111-y

Abstract

We give a new approach for the estimations of the eigenvalues of non-self-adjoint Sturm–Liouville operators with periodic and antiperiodic boundary conditions. Moreover, we give error estimations, and finally we present some numerical examples.

Highlights

Introduction and preliminary facts LetLk(q) for k = 0, 1 be the operators generated in L2[0, 1] by the differential expression –y + q(x)y and the boundary conditions y(1) = eiπky(0), y (1) = eiπky (0), (1)that is, periodic and antiperiodic boundary conditions, where q is a complex-valued summable function on [0, 1]
The first result in terms of the Fourier coefficients of the potential q was obtained by Dernek and Veliev [6], and this result was essentially improved by Shkalikov and Veliev [19]
One of the interesting approaches was given by Dinibütün and Veliev [7]. They considered the matrix form of the operator T(p) generated in L2[0, 1] by the differential expression –y + q(x)y and the boundary conditions y(2π) = y(0), y (2π) = y (0), where the potential q is in the form q(x) = p(x) + n:|n|>s qneinx and p(x) = n:|n|≤s qneinx, and they gave an approximation with very small errors for the eigenvalues of the periodic

Summary

Introduction

There are many studies about the numerical estimations of the small eigenvalues of the Sturm–Liouville operators with periodic and antiperiodic boundary conditions. They considered the matrix form of the operator T(p) generated in L2[0, 1] by the differential expression –y + q(x)y and the boundary conditions y(2π) = y(0), y (2π) = y (0), where the potential q is in the form q(x) = p(x) + n:|n|>s qneinx and p(x) = n:|n|≤s qneinx, and they gave an approximation with very small errors for the eigenvalues of the periodic

Results

Conclusion