On the Estimations of the Small Periodic Eigenvalues

Abstract

We estimate the small periodic and semiperiodic eigenvalues of Hill's operator with sufficiently differentiable potential by two different methods. Then using it we give the high precision approximations for the length of th gap in the spectrum of Hill-Sehrodinger operator and for the length of th instability interval of Hill's equation for small values of Finally we illustrate and compare the results obtained by two different ways for some examples.

An estimation of the eigenfunctions of periodic Sturm-Liouville problems

“…where the potential q x ð Þ ¼ P 1 k¼À1 e i2kx ¼ 2 cos 2x: The eigenvalue k can be taken from Dinibutun and Veliev (2013) which gives high precision results for the calculation of the small eigenvalues.…”

“…The approximation of order 10 À18 , 10 À15 and 10 À12 for the first 201 eigenvalues has been calculated and proved with the examples. This paper can be considered as a continuation of the paper (Dinibutun & Veliev, 2013).…”

“…boundary conditions, where q is a real periodic function with period p: The spectrum of the operators P(q) and S(q) consists of the eigenvalues called as periodic and semiperiodic eigenvalues respectively (Eastham, 1973). This paper gives the estimations for the eigenfunctions corresponding to the periodic and semiperiodic eigenvalues which have been calculated in Dinibutun and Veliev (2013) when the real periodic potential q belongs to the Sobolev space W l 1 0, p ½ with l > 1: These assumptions on the potential q imply that…”

“…There are many papers about numerical estimation of the periodic eigenvalues. However, in Dinibutun and Veliev (2013), the matrix form of the operator P(q) has been considered and obtained very sharp estimation for the small eigenvalues. The approximation of order 10 À18 , 10 À15 and 10 À12 for the first 201 eigenvalues has been calculated and proved with the examples.…”

On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions

“…Malathi et al [15] used the shooting method and direct integration method for computing eigenvalues of the periodic Sturm-Liouville problems. One of the interesting approaches was given by Dinibütün and Veliev [7]. They considered the matrix form of the operator T(p) generated in L 2 [0, 1] by the differential expression -y + q(x)y and the boundary conditions y(2π) = y(0), y (2π) = y (0),…”