Titchmarsh's asymptotic formula for periodic eigenvalues and an extension to the p-Laplacian

Abstract

An asymptotic formula for periodic eigenvalues, due to Titchmarsh, is taken to a higher degree of accuracy using a Prüfer transformation method, not depending on Floquet theory. With the idea of the rotation number, this method is then extended to the p-Laplacian.

On the asymptotic simplicity of periodic eigenvalues and Titchmarsh's formula

“…Note that the O-term in (1.3) was essentially given as O(m −1 ), . The present work was stimulated by the papers [7,8,9].…”

“…In the classical investigations, the order of the asymptotic estimates for the two eigenvalues λ 2m+1 , λ 2m+2 is closely related to the order of smoothness of the potential q. We mention in particular [1,3,5,6] and the latest results in [7]. In [1], Theorem 4.2.4, if, for r ≥ 1, q has an absolutely continuous (r−1)st derivative on (−∞, ∞), the asymptotic estimates for the λ 2m+1 , λ 2m+2 as m → ∞ have an error term of order o(m −r ).…”

“…Note that the O-term in (1.3) was essentially given as O(m −1 ), but Eastham [1] explained, Theorem 4.4.1, that Titchmarsh's proof produces only the O-term as in (1.3). For the first time, by using the Prüfer transformation, Brown and Eastham [7] proved, Theorem 2.2, that if q is locally integrable on (−∞, ∞), then the Titchmarsh's formula (1.3) can indeed be improved to the O-term O(m −1 ). The present work was stimulated by the papers [7,8,9].…”

“…For the first time, by using the Prüfer transformation, Brown and Eastham [7] proved, Theorem 2.2, that if q is locally integrable on (−∞, ∞), then the Titchmarsh's formula (1.3) can indeed be improved to the O-term O(m −1 ). The present work was stimulated by the papers [7,8,9].…”

A further note on the inverse nodal problem and Ambarzumyan problem for the p-Laplacian

“…In 2008, Brown and Eastham [4] derived a sharp asymptotic expansion of eigenvalues of the p-Laplacian with locally integrable and absolutely continuous (r − 1) derivative potentials respectively. Below is a version of their theorem for periodic eigenvalues of the p-Laplacian (1.1), (1.2).…”