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 $n$th gap in the spectrum of Hill-Sehrodinger 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.