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On the estimations of the small periodic eigenvalues. (English) Zbl 1303.34071

Summary: 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 \(n\)th gap in the spectrum of Hill-Schrödinger operator and for the length of \(n\)th instability interval of Hill’s equation for small values of \(n\). Finally we illustrate and compare the results obtained by two different ways for some examples.

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B09 Boundary eigenvalue problems for ordinary differential equations
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
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