On the Estimations of the Small Periodic Eigenvalues

Seza Dinibütün,O A Veliev

doi:10.1155/2013/145967

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.

Highlights

Let P(q) and S(q) be the operators generated in L2[0, π] by the differential expression−y󸀠󸀠 (x) + q (x) y (x) (1)with the periodic y (π) = y (0), y󸀠 (π) = y󸀠 (0) (2)and semiperiodic y (π) = −y (0), y󸀠 (π) = −y󸀠 (0) (3)boundary conditions, respectively, where q is a real periodic function with period π
We obtain an approximation of the eigenvalues λ±n(p) for n > ms, where m is the positive integer for determination of the error in estimations, by using the method of the paper [11], where the asymptotic formulas for the eigenvalues and eigenfunctions of the t-periodic boundary value problems were obtained. Using it and considering the matrix form of T(p) we give an approximation with very small errors for all small periodic and semiperiodic eigenvalues. We apply these investigations to get approximations order 10−18, 10−15, and 10−12 for the first 201 eigenvalues of the operator T with potentials p1(x) = 2 cos 2x, p2(x) = 2 cos 2x + 2 cos 4x, and p3(x) = 2 cos 2x + 2 cos 4x + 2 cos 6x, respectively, and give a comparison between the approximated eigenvalues obtained by the different ways
For i = 1, 2, 3, . . . hold, where n satisfies (41), Kn is defined in Theorem 2, and

Summary

Introduction

Let P(q) and S(q) be the operators generated in L2[0, π] by the differential expression. The spectrum of the operator L(q) generated in L2(−∞, ∞) by (1) consists of the intervals [λn−1(q), λ−n(q)] for n = 1, 2,. The estimations of the periodic and semiperiodic eigenvalues are the investigations of the spectrum of L(q) and of the stable intervals of (7). We apply these investigations to get approximations order 10−18, 10−15, and 10−12 for the first 201 eigenvalues of the operator T with potentials p1(x) = 2 cos 2x, p2(x) = 2 cos 2x + 2 cos 4x, and p3(x) = 2 cos 2x + 2 cos 4x + 2 cos 6x, respectively, and give a comparison between the approximated eigenvalues obtained by the different ways

On Applications of the Asymptotic Methods

Estimation of the Small Eigenvalues

Examples and Conclusion