Some inverse results for Hill's equation (Q1857013)

The author deals with an inverse problem for the Hill equation \[ y''(t)+ (\lambda- q(t)) y(t)= 0\tag{\(*\)} \] on \(\mathbb{R}\), where \(\lambda\) is a real parameter and \(q\) is a real-valued, locally integrable, periodic function of period \(\pi\). Denote by \(l_n\) the length of the \(n\)th instability interval of this equation. Let \(\{a^k_n\}^\infty_{n=0}\) and \(\{b^k_n\}^\infty_{n=1}\) be the cosine and sine Fourier coefficients of \(q^{(k)}\), \(k= 0,1,2,\dots\), respectively. The author proves \hskip 15mm (1) If \(l_m={\mathcal O}(n^{-1})\), then \(a_n, b_n={\mathcal O}(n^{-1})\) as \(n\to\infty\). \hskip 15mm (2) If \(l_n={\mathcal O}(n^{-(k+2)})\), then \(a^k_n, b^k_n={\mathcal O}(n^{-2})\) as \(n\to\infty\) and hence \(q^{(k)}\) is absolutely continuous. In the proof of these results, the following technics are employed. Thanks to the theorem by \textit{H. Hochstadt} [Arch. Ration. Mech. Anal. 19, 353--362 (1965; Zbl 0128.31201)], the problem reduces to the study of the Dirichlet eigenvalues of \((*)\) on shifted intervals \([\tau,\tau+ \pi]\), \(\tau\in [0,\pi)\). Using the Prüfer transformation, the author obtains a leading asymptotic expansion of such Dirichlet eigenvalues and apply that to prove (1). The proof of (2) is based on the author's and \textit{B. J. Harris'} earlier result [Proc. R. Soc. Edinb., Sect. A, Math. 130, 991--998 (2000; Zbl 0977.34077)] on the higher-order asymptotic expansion, which was demonstrated by analyzing the associated Riccati equation.