\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 41, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/41\hfil Inverse problems] {Inverse problems associated with the Hill operator} \author[A. A. Kira\c{c} \hfil EJDE-2016/41\hfilneg] {Alp Arslan Kira\c{c}} \address{Alp Arslan Kira\c{c} \newline Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University, 20070, Denizli, Turkey} \email{aakirac@pau.edu.tr} \thanks{Submitted April 29, 2015. Published January 27, 2016.} \subjclass[2010]{34A55, 34B30, 34L05, 47E05, 34B09} \keywords{Hill operator; inverse spectral theory; eigenvalue asymptotics; \hfill\break\indent Fourier coefficients} \begin{abstract} Let $\ell_n$ be the length of the $n$-th instability interval of the Hill operator $Ly=-y''+q(x)y$. We prove that if $\ell_n=o(n^{-2})$ and the set $\{(n\pi)^2: n \text{ is even and } n>n_0\}$ is a subset of the periodic spectrum of the Hill operator, then $q=0$ a.e., where $n_0$ is a sufficiently large positive integer such that $\ell_n<\varepsilon n^{-2}$ for all $n>n_0(\varepsilon)$ with some $\varepsilon>0$. A similar result holds for the anti-periodic case. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Consider the Hill operator \begin{equation} \label{1} Ly=-y''+q(x)y, \end{equation} where $q(x)$ is a real-valued summable function on $[0,1]$ and $q(x+1)=q(x)$. Let $\lambda_n$ and $\mu_n$ $(n=0,1,\ldots)$ denote, respectively, the $n$-th periodic and anti-periodic eigenvalues of the Hill operator \eqref{1} on $[0,1]$ with the periodic boundary conditions \begin{equation}\label{per.bc} y(0)=y(1),\quad y'(0)=y'(1), \end{equation} and the anti-periodic boundary conditions \[ y(0)=-y(1),\quad y'(0)=-y'(1). \] It is well-known \cite{Coddington:Levinson,Eastham} that \[ \lambda_0<\mu_0\leq \mu_1<\lambda_1\leq\lambda_2<\mu_2% \leq \mu_3<\dots \to \infty. \] The intervals $(\mu_{2m},\mu_{2m+1})$ and $(\lambda_{2m+1},\lambda_{2m+2})$ are respectively referred to as the $(2m+1)$-th and $(2m+2)$-th finite instability intervals of the operator $L$, while $(-\infty,\lambda_0)$ is called the zero-th instability interval. The length of the $n$-th instability interval of \eqref{1} will be denoted by $\ell_n$ ($n=2m+1,\,2m+2$). For further background see \cite{Magnus-Winkler, Marchenko, Hochstadt:determination}. Borg \cite{Borg}, Ungar \cite{Ungar} and Hochstadt \cite{Hochstadt:determination} proved independently of each other the following statement: \begin{quote} If $q(x)$ is real and integrable, and if all finite instability intervals vanish then $q(x)=0$ a.e. \end{quote} Hochstadt \cite{Hochstadt:determination} showed that if precisely one of the finite instability intervals does not vanish, then $q(x)$ is the elliptic function which satisfies \[ q''=3q^2+Aq+B \quad \text{a.e.}, \] where $A$ and $B$ are suitable constants. Hochstadt \cite{Hochstadt:determination} also proved that $q(x)$ is infinitely differentiable a.e. when $n$ finite instability intervals fail to vanish. For more results see \cite{Goldberg:determination, Goldberg:necessary, Goldberghoch:selected, Goldberghoch:finitenumber}. Furthermore, Hochstadt \cite{Hochstadt:Stability-Estimate} proved that the lengths of the instability intervals $\ell_n$ vanish faster than any power of $(1/n)$ for $q$ in $C_1^{\infty}$. McKean and Trubowitz \cite{McKean} established the converse: if $q$ is in $L_1^2$, the space of 1-periodic square integrable functions in $[0,1]$, and the length of the $n$-th instability interval for $n\geq 1$ is rapidly decreasing, then $q$ is in $C_1^{\infty}$. Later Trubowitz \cite{Trubowitz} proved the following: if $q$ is real analytic, the lengths of the instability intervals are exponentially decreasing. Conversely if $q$ is in $L_1^2$ and the lengths of the instability intervals are exponentially decreasing, $q$ is real analytic. Denoting the Fourier coefficients of q by \begin{equation} \label{cn} c_n=:(q, \exp(i2n\pi \cdot))_{L^2([0,1];dx)},\quad n\in \mathbb{N}\cup\{0\}, \end{equation} Coskun \cite{Coskun:invers} showed that \begin{equation}\label{coskun} \text{if $\ell_n=O(n^{-2})$, then $c_n=O(n^{-2})$ as $n\to \infty$}. \end{equation} At this point, we refer to some Ambarzumyan-type theorems in \cite{Ambarz, corri, Ambarzcoupled, anoteinver}. In 1929, Ambarzumyan \cite{Ambarz} obtained the following first theorem in inverse spectral theory: If $\{n^2 : n = 0,1,\ldots\}$ is the spectrum of the Sturm-Liouville operator \eqref{1} on $[0,1]$ with the Neumann boundary conditions, then $q = 0$ a.e. In \cite{corri}, they extended the classical Ambarzumyan's theorem for the Sturm-Liouville equation to the general separated boundary conditions, by imposing an additional condition on the potential function, and their result supplements the P{\"{o}}schel-Trubowitz inverse spectral theory (see \cite{Poschel}). In \cite{Ambarzcoupled}, based on the well-known extremal property of the first eigenvalue, they find two analogs of Ambarzumyan's theorem to the Sturm-Liouville systems of $n$ dimension under periodic or anti-periodic boundary conditions. In \cite{anoteinver}, by using the Rayleigh-Ritz inequality and imposing a condition on the second term in the Fourier cosine series (see \eqref{cond}), they proved the following Ambarzumyan-type theorem: \begin{itemize} \item[(a)] If all periodic eigenvalues of Hill's equation \eqref{1} are nonnegative and they include $\{(2m\pi)^2: m\in \mathbb{N}\}$, then $q=0$ a.e. \item[(b)] If all anti-periodic eigenvalues of Hill's equation \eqref{1} are not less than $\pi^2$ and they include $\{(2m-1)^2\pi^2: m\in \mathbb{N}\}$, and \begin{equation} \label{cond} \int_0^{1}q(x)\cos(2\pi x)\,dx\geq 0, \end{equation} then $q=0$ a.e. \end{itemize} Recently, in \cite{kırac:ambars}, we obtain the classical Ambarzumyan's theorem for the Sturm-Liouville operators with $q\in L^{1}[0,1]$ and quasi-periodic boundary conditions in cases when there is not any additional condition on the potential $q$ such as \eqref{cond}. In this paper, we prove the following inverse spectral result, more precisely, a uniqueness-type result of the following form: \begin{theorem} \label{main0} Denote the $n$th instability interval by $\ell_n$, and suppose that $\ell_n=o(n^{-2})$ as $n\to \infty$. Then the following two assertions hold: \begin{itemize} \item[(i)] If $\{(n\pi)^2: \text{$n$ even and $n>n_0$}\}$ is a subset of the periodic spectrum of the Hill operator then $q=0$ a.e. on $(0,1)$; \item[(ii)] If $\{(n\pi)^2: \text{$n$ odd and $n>n_0$}\}$ is a subset of the anti-periodic spectrum of the Hill operator then $q=0$ a.e. on $(0,1)$. \end{itemize} Given $\varepsilon>0$, thee exists $n_0=n_0(\varepsilon)\in \mathbb{N}$, a sufficiently large positive integer such that \[ \ell_n<\varepsilon n^{-2}\quad \text{for all $n>n_0(\varepsilon)$.} \] \end{theorem} Theorem \ref{main0} is deduced from the following result. \begin{theorem} \label{thm1.2} Denote the Fourier coefficients of $q$ by $c_n$ (see \eqref{cn}), and assume $\ell_n=o(n^{-2})$. Then $c_n=o(n^{-2})$ as $n\to \infty$. \end{theorem} Note that, from Theorem \ref{thm1.2}, the assertion in \eqref{coskun} holds with the improved $o$-terms $o(n^{-2})$. In Ambarzumyan-type theorems, it is necessary to specify the whole spectrum. In \cite{FreilingYurko}, they proved that it is enough to know the first eigenvalue only. Unlike the above works, to prove of Theorem \ref{main0}, we have information only on the sufficiently large eigenvalues of the spectrum of the Hill operator. Also, the proof does not depend on multiplicities of the given eigenvalues. \section{Preliminaries and proof of main results}\label{asyl} We shall consider only the periodic (for even $n$) eigenvalues of the Hill operator. The anti-periodic (for odd $n$) problem is completely similar. It is well known \cite[Theorem 4.2.3]{Eastham} that the periodic eigenvalues $\lambda_{2m+1}, \lambda_{2m+2}$ are asymptotically located in pairs such that \begin{equation}\label{asy2} \lambda_{2m+1}=\lambda_{2m+2}+o(1)=(2m+2)^2\pi^2+o(1) \end{equation} for sufficiently large $m$. From this formula, for all $k\neq 0,(2m+2)$ and $k\in \mathbb{Z}$, the inequality \begin{equation} \label{dist1} |\lambda-(2(m-k)+2)^2\pi^2|>|k||(2m+2)-k|>C\,m, \end{equation} is satisfied by both eigenvalues $\lambda_{2m+1}$ and $\lambda_{2m+2}$ for large $m$, where, here and in subsequent relations, $C$ denotes a positive constant whose exact value is not essential. Note that, when $q=0$, the system $\{e^{-i(2m+2)\pi x}, e^{i(2m+2)\pi x}\}$ is a basis of the eigenspace corresponding to the double eigenvalues $(2m+2)^2\pi^2$ of the problem \eqref{1}-\eqref{per.bc}. To obtain the asymptotic formulas for the periodic eigenvalues $\lambda_{2m+1}, \lambda_{2m+2}$ corresponding to the normalized eigenfunctions $\Psi_{m,1}(x),\Psi_{m,2}(x)$ respectively, let us consider the well-known relation, for sufficiently large $m$, \begin{equation} \label{m1} \Lambda_{m,j,m-k}(\Psi_{m,j},e^{i(2(m-k)+2)\pi x})=(q\,\Psi_{m,j},e^{i(2(m-k)+2)\pi x}), \end{equation} where $\Lambda_{m,j,m-k}=(\lambda_{2m+j}-(2(m-k)+2)^2\pi^2)$, $j=1,2.$ The relation \eqref{m1} can be obtained from the equation \eqref{1} by multiplying $e^{i(2(m-k)+2)\pi x}$ and using integration by parts. From \cite[Lemma 1]{Melda.O}, to iterate \eqref{m1} for $k=0$, in the right hand-side of formula \eqref{m1}, we use the following relations \begin{gather} \label{m2} (q\,\Psi_{m,j},e^{i (2m+2)\pi x})=\sum_{m_1=-\infty}^{\infty}c_{m_1}(\Psi_{m,j},e^{i (2(m-m_1)+2)\pi x}), \\ \label{m3} |(q\,\Psi_{m,j},e^{i (2(m-m_1)+2)\pi x})|< 3M \end{gather} for all large $m$, where $j=1,2$ and $M=\sup_{m\in \mathbb{Z}}|c_{m}|$. First, we fix the terms with indices $m_1=0,(2m+2)$. Then all the other terms in the right hand-side of \eqref{m2} are replaced, in view of \eqref{dist1} and \eqref{m1} for $k=m_1$, by \[ c_{m_1}\frac{(q\,\Psi_{m,j},e^{i(2(m-m_1)+2)\pi x})}{\Lambda_{m,j,m-m_1}}. \] In the same way, by applying the above process for the eigenfunction $e^{-i(2m+2)\pi x}$ corresponding to the eigenvalue $(2m+2)^2\pi^2$ of the problem \eqref{1}-\eqref{per.bc} for $q=0$, we obtain the following lemma (see also Section 2 in \cite{kyrac:abstract, kyrac:titch}). \begin{lemma}\label{lemmaitera} The following relations hold for sufficiently large $m$: (i) \begin{equation}\label{m4123} [\Lambda_{m,j,m}- c_0-\sum_{i=1}^2a_i(\lambda_{2m+j})]u_{m,j} =[c_{2m+2}+\sum_{i=1}^2b_i(\lambda_{2m+j})]v_{m,j}+R_2, \end{equation} where $j=1,2$, \begin{gather} u_{m,j}=(\Psi_{m,j},e^{i(2m+2)\pi x}),\quad v_{m,j}=(\Psi_{m,j},e^{-i(2m+2)\pi x}), \nonumber\\ \label{a1a2} \begin{gathered} a_1(\lambda_{2m+j})=\sum_{m_1}\frac{c_{m_1}c_{-m_1}}{\Lambda_{m,j,m-m_1}},\\ a_2(\lambda_{2m+j})=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2} c_{-m_1-m_2}}{\Lambda_{m,j,m-m_1}\Lambda_{m,j,m-m_1-m_2}}, \end{gathered} \\ b_1(\lambda_{2m+j})=\sum_{m_1}\frac{c_{m_1}c_{2m+2-m_1}}{\Lambda_{m,j,m-m_1}}, \nonumber\\ b_2(\lambda_{2m+j})=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2}c_{2m+2-m_1-m_2}} {\Lambda_{m,j,m-m_1} \Lambda_{m,j,m-m_1-m_2}}, \nonumber \\ \label{R} R_2=\sum_{m_1,m_2,m_3}\frac{c_{m_1}c_{m_2}c_{m_3} (q\Psi_{m,j}(x),e^{i(2(m-m_1-m_2-m_3)+2)\pi x})}{\Lambda_{m,j,m-m_1}\, \Lambda_{m,j,m-m_1-m_2}\Lambda_{m,j,m-m_1-m_2-m_3}}. \nonumber \end{gather} The summations in these formulas are taken over all integers $m_1,m_2, m_3$ such that $m_1, m_1+m_2, m_1+m_2+m_3\neq 0,\,2m+2$. (ii) \begin{equation}\label{m412} [\Lambda_{m,j,m}-c_0-\sum_{i=1}^2 a'_i(\lambda_{2m+j})]v_{m,j} =[c_{-2m-2}+\sum_{i=1}^2b'_i(\lambda_{2m+j})]u_{m,j}+R'_2, \end{equation} where $j=1,2$, \begin{gather*} a'_1(\lambda_{2m+j})=\sum_{m_1}\frac{c_{m_1}c_{-m_1}}{\Lambda_{m,j,m+m_1}},\\ a'_2(\lambda_{2m+j})=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2}c_{-m_1-m_2}} {\Lambda_{m,j,m+m_1}\,\Lambda_{m,j,m+m_1+m_2}}, \\ b'_1(\lambda_{2m+j})=\sum_{m_1}\frac{c_{m_1}c_{-2m-2-m_1}}{\Lambda_{m,j,m+m_1}},\\ b'_2(\lambda_{2m+j})=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2}c_{-2m-2-m_1-m_2}} {\Lambda_{m,j,m+m_1}\,\Lambda_{m,j,m+m_1+m_2}}, \end{gather*} \begin{equation}\label{R'} R'_2=\sum_{m_1,m_2,m_3}\frac{c_{m_1}c_{m_2}c_{m_3} (q\Psi_{m,j}(x),e^{i(2(m+m_1+m_2+m_3)+2)\pi x})}{\Lambda_{m,j,m+m_1} \Lambda_{m,j,m+m_1+m_2}\,\Lambda_{m,j,m+m_1+m_2+m_3}} \end{equation} and the sums in these formulas are taken over all integers $m_1,m_2, m_3$ such that $m_1, m_1+m_2, m_1+m_2+m_3\neq 0,\,-2m-2$. \end{lemma} Note that, by substituting $m_1=-k_1$ and $m_1+m_2=-k_1$, $m_2=k_2$ into the relations $a'_1(\lambda_{2m+j})$ and $a'_2(\lambda_{2m+j})$ respectively, we have \begin{equation}\label{a1=a1} a_i(\lambda_{2m+j})=a'_i(\lambda_{2m+j})\quad \text{for $i=1,2$.} \end{equation} Here, using the equality \[ \frac{1}{m_1(2m+2-m_1)}=\frac{1}{2m+2}\Big(\frac{1}{m_1}+\frac{1}{2m+2-m_1}\Big), \] we obtain the relation \[ \sum_{m_1\neq 0,(2m+2)}\frac{1}{|m_1(2m+2-m_1)|}=O\Big(\frac{\ln|m|}{m}\Big). \] This, together with \eqref{dist1}, \eqref{m1} and \eqref{m3}, gives the following estimates (see, respectively, \eqref{R} and \eqref{R'} for $R_2$ and $R'_2$) \begin{equation}\label{m45} R_2,\; R'_2=O\Big(\big(\frac{\ln| m|}{m}\big)^3\Big). \end{equation} Moreover, in view of \eqref{dist1}, \eqref{m1} and \eqref{m3}, we obtain (see \cite[Theorem 2]{Melda.O}, \cite{kyrac:titch}) \begin{equation}\label{kare} \sum_{k\in \mathbb{Z};\,k\neq \pm(m+1)} \big|(\Psi_{m,j},e^{i2k\pi x})\big|^2=O\Big(\frac{1}{m^2}\Big). \end{equation} Therefore, the expansion of the normalized eigenfunctions $\Psi_{m,j}(x)$ by the orthonormal basis $\{e^{i2k\pi x}:k\in \mathbb{Z}\}$ on $[0,1]$ has the form \begin{equation}\label{m7} \Psi_{m,j}(x)=u_{m,j}\,e^{i(2m+2)\pi x}+v_{m,j}\,e^{-i(2m+2)\pi x}+h_{m}(x), \end{equation} where \begin{equation}\label{m8} \begin{gathered} (h_{m},e^{\mp i(2m+2)\pi x})=0,\quad \|h_{m}\|=O(m^{-1}),\\ \sup_{x\in[0,1]}|h_{m}(x)|=O\Big(\frac{\ln|m|}{m}\Big), \quad |u_{m,j}|^2+|v_{m,j}|^2=1+O\big(m^{-2}\big). \end{gathered} \end{equation} \subsection*{Proof of Theorem \ref{thm1.2}} First, let us estimate the expressions in \eqref{m4123} and \eqref{m412}. From \eqref{asy2}, \eqref{dist1} and \eqref{kare}, one can readily see that \begin{equation}\label{dif} \begin{aligned} &\sum_{m_1\neq 0,\pm(2m+2)} \big|\frac{1}{\Lambda_{m,j,m\mp m_1}}-\frac{1}{\Lambda_{m,0,m\mp m_1}}\big| \\ &\leq C|\Lambda_{m,j,m}|\sum_{m_1\neq 0,\pm(2m+2)}|m_1|^{-2}|2m+2\mp m_1|^{-2} =o\big(m^{-2}\big), \end{aligned} \end{equation} where $\Lambda_{m,0,m\mp m_1}=((2m+2)^2\pi^2-(2(m\mp m_1)+2)^2\pi^2)$. Thus, we obtain \begin{equation}\label{a1} a_i(\lambda_{2m+j})=a_i((2m+2)^2\pi^2)+o\big(m^{-2}\big) \quad\text{for } i=1,2. \end{equation} Here, by \eqref{dif}, we also have, arguing as in \cite[Lemma 3]{kyrac:titch} (see also \cite[Lemma 6]{Veliev:Shkalikov}), \begin{equation}\label{I0} \begin{aligned} b_1(\lambda_{2m+j}) &=\frac{1}{4\pi^2}\sum_{m_1\neq 0,(2m+2)} \frac{c_{m_1}c_{2m+2-m_1}}{m_1(2m+2-m_1)}+o\big(m^{-2}\big)\\ &=-\int_0^{1}(Q(x)-Q_0)^2\,e^{-i2(2m+2)\pi x}dx+o\big(m^{-2}\big) \\ &=\frac{-1}{i2\pi(2m+2)}\int_0^{1}2(Q(x)-Q_0)\,q(x)\,e^{-i2(2m+2)\pi x}dx +o\big(m^{-2}\big), \end{aligned} \end{equation} where \begin{equation} \label{Q0} \begin{gathered} Q(x)-Q_0=\sum_{m_1\neq 0}Q_{m_1}\,e^{i2m_1\pi x}, \\ Q_{m_1}=:(Q(x),e^{i2m_1\pi x})=\frac{c_{m_1}}{i2\pi m_1}, \quad m_1\neq 0, \end{gathered} \end{equation} are the Fourier coefficients with respect to the system $\{e^{i2m_1\pi x}: m_1\in\mathbb{Z}\}$ of the function $Q(x)=\int_0^{x}q(t)\, dt.$ For the proof of Theorem \ref{thm1.2}, we suppose without loss of generality that $c_0=0$, so that $Q(1)=c_0=0$. Now using the assumption $\ell_n=o(n^{-2})$ of the theorem it is also true that $\ell_n=O(n^{-2})$. In view of \eqref{coskun}, we obtain $c_n=O(n^{-2})$ as $n\to\infty$. Thus, from \cite[Lemma 5]{Hochstadt:determination}, we obtain that $q(x)$ is absolutely continuous a.e. Hence integration by parts, together with $Q(1)=0$, gives \begin{equation} \begin{aligned} &b_1(\lambda_{2m+j})\\ & =\frac{1}{2\pi^2(2m+2)^2}\int_0^{1}\left(q^2(x)+(Q(x)-Q_0)q'(x) \right)e^{-i2(2m+2)\pi x}dx +o\big(m^{-2}\big). \end{aligned} \end{equation} Since $q(x)$ is absolutely continuous a.e., $\left(q^2(x)+(Q(x)-Q_0)q'(x)\right)\in L^{1}[0,1]$. By the Riemann-Lebesgue lemma, we find that \begin{equation}\label{b1o} b_1(\lambda_{2m+j})=o\big(m^{-2}\big). \end{equation} Similarly \begin{equation}\label{b1oprime} b'_1(\lambda_{2m+j})=o\big(m^{-2}\big). \end{equation} Let us prove that \begin{equation}\label{b2oandprime} b_2(\lambda_{2m+j}),\; b'_2(\lambda_{2m+j})=o\big(m^{-2}\big). \end{equation} Taking into account that $q(x)$ is absolutely continuous a.e. and periodic, we obtain $c_{m_1}c_{m_2}c_{\pm(2m+2)-m_1-m_2}=o\left(m^{-1}\right)$ (see \cite[p. 665]{Veliev:Shkalikov}). Using this and arguing as in the proof of \eqref{m45}, we obtain \begin{align*} |b_2(\lambda_{2m+j})| &=o\left(m^{-1}\right) \sum_{m_1,m_2}\frac{1}{|m_1(2m+2-m_1)(m_1+m_2)(2m+2-m_1-m_2)|} \\ &=o\left(m^{-1}\right)O\Big(\big(\frac{\ln| m|}{m}\big)^2\Big)=o\big(m^{-2}\big). \end{align*} Thus, the first estimate of \eqref{b2oandprime} is proved. Similarly $b'_2(\lambda_{2m+j})=o\big(m^{-2}\big)$. Substituting the estimates given by \eqref{a1=a1}, \eqref{m45}, \eqref{a1} and \eqref{b1o}-\eqref{b2oandprime} into the relations \eqref{m4123} and \eqref{m412}, we find that \begin{gather}\label{son} [\Lambda_{m,j,m}-\sum_{i=1}^2a_i((2m+2)^2\pi^2)]u_{m,j} =c_{2m+2}v_{m,j}+o\big(m^{-2}\big), \\ \label{son'} [\Lambda_{m,j,m}-\sum_{i=1}^2a_i((2m+2)^2\pi^2)]v_{m,j} =c_{-2m-2}\,u_{m,j}+o\big(m^{-2}\big) \end{gather} for $j=1,2$. Now suppose that, contrary to what we want to prove, there exists an increasing sequence $\{m_{k}\}$ $(k=1,2,\ldots)$ such that \begin{equation}\label{mk} |c_{2m_{k}+2}|>C m_{k}^{-2}\quad\text{for some $C>0$}. \end{equation} Further, the formula \eqref{m8} for $m=m_{k}$ implies that either $|u_{m_{k},j}|>1/2$ or $|v_{m_{k},j}|>1/2$ for sufficiently large $m_{k}$. Without loss of generality, we assume that $|u_{m_{k},j}|>1/2$. Then it follows from both \eqref{son} and \eqref{son'} for $m=m_{k}$ that \begin{equation}\label{sameo} [\Lambda_{m_{k},j,m_{k}}-\sum_{i=1}^2a_i((2m_{k}+2)^2\pi^2)]\sim c_{2m_{k}+2}, \end{equation} where the notation $a_{m}\sim b_{m}$ means that there exist constants $c_1$, $c_2$ such that $01/2$, implies that \begin{equation}\label{vsimilar} u_{m_{k},j}\sim v_{m_{k},j}\sim 1. \end{equation} Multiplying \eqref{son'} for $m=m_{k}$ by $c_{2m_{k}+2}$, and then using \eqref{son} for $m=m_{k}$ in \eqref{son'}, we arrive at the relation \begin{equation} \begin{aligned} &[\Lambda_{m_{k},j,m_{k}}-\sum_{i=1}^2 a_i((2m_{k}+2)^2\pi^2)] \Big([\Lambda_{m_{k},j,m_{k}} \\ & -\sum_{i=1}^2a_i((2m_{k}+2)^2\pi^2)]u_{m_{k},j}+o\big(m_{k}^{-2}\big) \Big)\\ &=|c_{2m_{k}+2}|^2\,u_{m_{k},j}+c_{2m_{k}+2}\, o\big(m_{k}^{-2}\big), \end{aligned} \end{equation} which, by \eqref{sameo} and \eqref{vsimilar}, implies \begin{equation}\label{final} \Lambda_{m_{k},j,m_{k}}-\sum_{i=1}^2a_i((2m_{k}+2)^2\pi^2) =\pm|c_{2m_{k}+2}|+o\big(m_{k}^{-2}\big) \end{equation} for $j=1,2$. Let us prove that the periodic eigenvalues for large $m_{k}$ are simple. Assume that there exist two orthogonal eigenfunctions $\Psi_{m_{k},1}(x)$ and $\Psi_{m_{k},2}(x)$ corresponding to $\lambda_{2m_{k}+1}=\lambda_{2m_{k}+2}$. From the argument of \cite[Lemma 4]{Veliev:Shkalikov}, using the relation \eqref{m7} with $\|h_{m_{k}}\|=O(m_{k}^{-1})$ for the eigenfunctions $\Psi_{m_{k},j}(x)$ and the orthogonality of eigenfunctions, we can choose these eigenfunctions such that either $u_{m_{k},j}=0$ or $v_{m_{k},j}=0$, which contradicts \eqref{vsimilar}. Since the eigenfunctions $\Psi_{m_{k},1}$ and $\overline{\Psi_{m_{k},2}}$ of the self-adjoint problem corresponding to the different eigenvalues $\lambda_{2m_{k}+1}\neq\lambda_{2m_{k}+2}$ are orthogonal we find, by \eqref{m7}, that \begin{equation}\label{orteigen} 0=(\Psi_{m_{k},1},\overline{\Psi_{m_{k},2}}) =u_{m_{k},2}v_{m_{k},1}+u_{m_{k},1}v_{m_{k},2}+O(m_{k}^{-1}). \end{equation} Note that, for the simple eigenvalues in \eqref{final}, there are two cases. First case: The simple eigenvalues $\lambda_{2m_{k}+1}$ and $\lambda_{2m_{k}+2}$ in \eqref{final} corresponds respectively to the lower sign $-$ and upper sign $+$. Then \[ \ell_{2{m_{k}}+2}=\lambda_{m_{k},2,m_{k}}-\lambda_{m_{k},1,m_{k}} =2|c_{2m_{k}+2}|+o\big(m_{k}^{-2}\big), \] which implies that (see \eqref{mk}) $\ell_{2{m_{k}}+2}>C m_{k}^{-2}$ for some $C$. This contradicts the hypothesis that $\ell_{2{m_{k}}+2}=o(m_{k}^{-2})$. Now let us consider the second case: We assume that both the simple eigenvalues correspond to the lower sign $-$ (the proof corresponding to the upper sign $+$ is similar). Then $\Lambda_{m_{k},2,m_{k}}-\Lambda_{m_{k},1,m_{k}}=o\big(m_{k}^{-2}\big)$. Using this, \eqref{son} and \eqref{final}, we have \begin{gather}\label{fark1} o\big(m_{k}^{-2}\big) u_{m_{k},2} =c_{2m_{k}+2} v_{m_{k},2}+|c_{2m_{k}+2}| u_{m_{k},2}+o\big(m_{k}^{-2}\big), \\ \label{fark2} o\big(m_{k}^{-2}\big) u_{m_{k},1} =-c_{2m_{k}+2} v_{m_{k},1}-|c_{2m_{k}+2}| u_{m_{k},1}+o\big(m_{k}^{-2}\big). \end{gather} Therefore, multiplying both sides of \eqref{fark1} and \eqref{fark2} by $v_{m_{k},1}$ and $v_{m_{k},2}$ respectively and adding the results, we have, in view of \eqref{mk}, \[ u_{m_{k},2}v_{m_{k},1}-u_{m_{k},1}v_{m_{k},2}=o(1). \] This, together with \eqref{orteigen}, gives $u_{m_{k},2}v_{m_{k},1}=o(1)$ which contradicts \eqref{vsimilar}. Thus the assumption \eqref{mk} is false, that is, $c_{2m+2}=o\big(m^{-2}\big)$. A similar result holds for the anti-periodic problem, that is, $c_{2m+1}=o\big(m^{-2}\big)$. The theorem is proved. For the proof of Theorem \ref{main0}, we need the sharper estimates in the following lemma. \begin{lemma}\label{lemma12} Let $q(x)$ be absolutely continuous a.e. and $c_0=0$. Then, for all sufficiently large $m$, we have the following estimates \begin{equation}\label{lemma1} \begin{gathered} a_1(\lambda_{2m+j})=\frac{-1}{(2\pi(2m+2))^2}\int_0^{1}q^2(x)dx+o\big(m^{-2}\big), \\ a_2(\lambda_{2m+j})=o\big(m^{-2}\big). \end{gathered} \end{equation} \end{lemma} \begin{proof} First, let us consider $a_1(\lambda_{2m+j})$. By \eqref{dif} we obtain \[ a_1(\lambda_{2m+j})=\frac{1}{4\pi^2}\sum_{m_1\neq 0,(2m+2)} \frac{c_{m_1}c_{-m_1}}{m_1(2m+2-m_1)}+o\big(m^{-2}\big). \] Arguing as in \cite[Lemma 3]{kyrac:titch} (see also \cite[Lemma 2.3(a)]{veliev;arşiv}), we obtain, in our notation, \begin{equation}\label{d3} \begin{aligned} &a_1(\lambda_{2m+j})\\ &=\frac{1}{2\pi^2}\sum_{m_1> 0,m_1\neq(2m+2)}\frac{c_{m_1}c_{-m_1}} {(2m+2+m_1)(2m+2-m_1)}+o\big(m^{-2}\big) \\ &=\int_0^{1}(G^{+}(x,m)-G^{+}_0(m))^2\,e^{i2(4m+4)\pi x}\,dx +o\big(m^{-2}\big) \\ &= \frac{-2}{i2\pi(4m+4)}\int_0^{1}\big(G^{+}(x,m)-G^{+}_0(m)\big)\\ &\quad \times \big(q(x)e^{-i2(2m+2)\pi x} -c_{2m+2}\big)e^{i2(4m+4)\pi x}dx +o\big(m^{-2}\big) \\ \end{aligned} \end{equation} where \begin{equation}\label{d4} G^{\pm}_{m_1}(m)=:(G^{\pm}(x,m), e^{i2m_1\pi x}) =\frac{c_{m_1\pm(2m+2)}}{i2\pi m_1}, \end{equation} for $m_1\neq 0$, are the Fourier coefficients with respect to $\{e^{i2m_1\pi x}: m_1\in\mathbb{Z}\}$ of the functions \begin{equation}\label{d2} G^{\pm}(x,m)=\int_0^{x}q(t)\,e^{\mp i2(2m+2)\pi t}dt-c_{\pm(2m+2)}x \end{equation} and \[ G^{\pm}(x,m)-G^{\pm}_0(m)=\sum_{m_1\neq(2m+2)} \frac{c_{m_1}}{i2\pi(m_1\mp(2m+2))}\,e^{i2(m_1\mp(2m+2))\pi x}. \] Here, taking into account the \cite[Lemma 1]{kyrac:titch} and \eqref{d2}, we have the estimate \begin{equation}\label{ggg} G^{\pm}(x,m)-G^{\pm}_0(m)=G^{\pm}(x,m)-\int_0^{1}G^{\pm}(x,m)\, dx=o(1) \quad\text{as $m\to\infty$} \end{equation} uniformly in $x$. From the equalities (see \eqref{d2}) \begin{equation}\label{gg} G^{\pm}(1,m)=G^{\pm}(0,m)=0, \end{equation} and since $q(x)$ is absolutely continuous a.e., integration by parts gives, for the right hand-side of \eqref{d3}, the value \begin{align*} a_1(\lambda_{2m+j}) &=\frac{-1}{(2\pi(2m+2))^2} \Big[\int_0^{1}q^2+\int_0^{1}(G^{+}(x,m)-G^{+}_0(m))q'(x) e^{i2(2m+2)\pi x}dx\Big] \\ &\quad +\frac{|c_{2m+2}|^2}{(2\pi(2m+2))^2}+o\big(m^{-2}\big) \end{align*} for sufficiently large $m$. Thus, by using the Riemann-Lebesgue lemma, this with $(G^{+}(x,m)-G^{+}_0(m))q'(x)\in L^{1}[0,1]$ implies the first equality of \eqref{lemma1}. Now, it remains to prove that $a_2(\lambda_{2m+j})=o\big(m^{-2}\big)$. Similarly, by \eqref{a1} for $i=2$, we obtain \begin{equation}\label{a2} \begin{aligned} a_2(\lambda_{2m+j}) &=\sum_{m_1,m_2}\frac{(2\pi)^{-4}\,c_{m_1}c_{m_2}c_{-m_1-m_2}}{m_1(2m+2-m_1) (m_1+m_2)(2m+2-m_1-m_2)}\\ &\quad +o\big(m^{-2}\big). \end{aligned} \end{equation} As in \cite[Lemma 4]{kyrac:titch}, using the summation variable $m_2$ to represent the previous $m_1+m_2$ in \eqref{a2}, we write \eqref{a2} in the form \[ a_2(\lambda_{2m+j})=\frac{1}{(2\pi)^{4}}\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2-m_1} c_{-m_2}}{m_1(2m+2-m_1)m_2(2m+2-m_2)}+o\big(m^{-2}\big), \] where the forbidden indices in the sums take the form of $m_1, m_2\neq 0,\,2m+2$. Here the equality \[ \frac{1}{k(2m+2-k)}=\frac{1}{2m+2}\Big(\frac{1}{k}+\frac{1}{2m+2-k}\Big) \] gives \begin{equation}\label{equal1} a_2(\lambda_{2m+j})=\frac{1}{(2\pi)^{4}(2m+2)^2}\sum_{j=1}^{4}S_{j}, \end{equation} where \begin{gather*} S_1=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2-m_1}c_{-m_2}}{m_1m_2},\quad S_2=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2-m_1}c_{-m_2}}{m_2(2m+2-m_1)}, \\ S_3=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2-m_1}c_{-m_2}}{m_1(2m+2-m_2)},\quad S_4=\sum_{m_1,m_2}\frac{c_{m_1}c_{m_2-m_1}c_{-m_2}}{(2m+2-m_1)(2m+2-m_2)}. \end{gather*} Using \eqref{Q0}, integration by parts and the assumption $c_0=0$ which implies $Q(1)=0$, we deduce that \begin{equation}\label{s11} S_1=4\pi^2\int_0^{1}(Q(x)-Q_0)^2q(x)\,dx=0. \end{equation} Similarly, in view of \eqref{Q0} and \eqref{d4}-\eqref{gg}, we obtain, by the Riemann-Lebesgue lemma, \begin{gather*} S_2=-4\pi^2\int_0^{1}(Q(x)-Q_0)(G^{+}(x,m)-G^{+}_0(m)) q(x)e^{i2(2m+2)\pi x}dx=o(1), \\ S_3=-4\pi^2\int_0^{1}(Q(x)-Q_0)(G^{-}(x,m)-G^{-}_0(m)) q(x)e^{-i2(2m+2)\pi x}dx=o(1) \end{gather*} and, by \eqref{ggg}, \[ S_4=4\pi^2\int_0^{1}(G^{+}(x,m)-G^{+}_0(m))(G^{-}(x,m) -G^{-}_0(m))\,q(x)\,dx=o(1). \] Thus, \eqref{equal1} implies that $a_2(\lambda_{2m+j})=o\big(m^{-2}\big)$. The proof is complete. \end{proof} \subsection*{Proof of Theorem \ref{main0}} (i) First let us prove that $c_0=0$. Considering the first step of the procedure in Lemma \ref{lemmaitera}, and using the estimate in \eqref{m45}, we may rewrite the relations \eqref{m4123} and \eqref{m412} as follows: \begin{equation}\label{sonnnnn} \begin{gathered} [\Lambda_{m,j,m}- c_0]u_{m,j}=c_{2m+2}v_{m,j} +O\Big(\frac{\ln| m|}{m}\Big), \\ [\Lambda_{m,j,m}- c_0]v_{m,j}=c_{-2m-2}\,u_{m,j} +O\Big(\frac{\ln| m|}{m}\Big) \end{gathered} \end{equation} for $j=1,2$ and sufficiently large $m$. By using the assumption $\ell_{2m+2}=o(m^{-2})$, namely, $\ell_n=o(n^{-2})$ for even $n=2m+2$ and Theorem \ref{thm1.2} which implies $c_{\mp(2m+2)}=o(m^{-2})$, we obtain the relations (see \eqref{sonnnnn}) in the form \begin{gather}\label{c01} [\Lambda_{m,j,m}- c_0]u_{m,j}=O\big(\frac{\ln| m|}{m}\big), \\ \label{c02} [\Lambda_{m,j,m}- c_0]v_{m,j}=O\big(\frac{\ln| m|}{m}\big). \end{gather} Again by \eqref{m8}, we have either $|u_{m,j}|>1/2$ or $|v_{m,j}|>1/2$ for large $m$. In either case, in view of \eqref{c01} and \eqref{c02}, there exists a sufficiently large positive integer $N_0$ such that both the eigenvalues $\lambda_{2m+j}$ (see definition of \eqref{m1}) satisfy the estimate \begin{equation}\label{c0son} \lambda_{2m+j}=(2m+2)^2\pi^2+c_0+O\big(\frac{\ln| m|}{m}\big) \end{equation} for all $m>N_0$ and $j=1,2$. Under the assumption of Theorem \ref{main0} (i), when $m>\max\{(n_0-2)/2,N_0\}$, the eigenvalue $(2m+2)^2\pi^2$ corresponds to the eigenvalue $\lambda_{2m+1}$ or $\lambda_{2m+2}$. In either case we obtain $c_0=0$ by \eqref{c0son}. Finally, for sufficiently large $m$, substituting the estimates of $a_i(\lambda_{2m+j})$, $a'_i(\lambda_{2m+j})$, $b_i(\lambda_{2m+j})$, $b'_i(\lambda_{2m+j})$, $R_2$, $R'_2$ for $i=1,2$, given by Lemma \ref{lemma12} with the equalities $a_i(\lambda_{2m+j})=a'_i(\lambda_{2m+j})$ (see \eqref{a1=a1}), \eqref{b1o}-\eqref{b2oandprime} and \eqref{m45} into the relations \eqref{m4123} and \eqref{m412} and using $c_0=0$, we find the relations in the form \begin{equation}\label{q2son} \begin{gathered} \Big[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^2}\int_0^{1}q^2\Big]u_{m,j} =c_{2m+2}v_{m,j}+o\big(m^{-2}\big), \\ \Big[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^2}\int_0^{1}q^2\Big]v_{m,j} =c_{-2m-2}\,u_{m,j}+o\big(m^{-2}\big) \end{gathered} \end{equation} for $j=1,2$. In the same way, by using the assumption $\ell_{2m+2}=o(m^{-2})$ and Theorem \ref{thm1.2}, we write \eqref{q2son} in the form \begin{gather*} \Big[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^2}\int_0^{1}q^2\Big]u_{m,j} =o\big(m^{-2}\big), \\ \Big[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^2}\int_0^{1}q^2\Big]v_{m,j} =o\big(m^{-2}\big). \end{gather*} Thus, arguing as in the proof of \eqref{c0son}, there exists a positive large number $N_1$ such that the eigenvalues $\lambda_{2m+j}$ satisfy the following estimate \begin{equation}\label{sssson} \lambda_{2m+j}=(2m+2)^2\pi^2-\frac{1}{(2\pi(2m+2))^2} \int_0^{1}q^2+o\big(m^{-2}\big) \end{equation} for all $m>N_1$ and $j=1,2$. Let $m>\max\{(n_0-2)/2,N_1\}$. Using the same argument as above, by \eqref{sssson}, we obtain $\int_0^{1}q^2=0$ which implies that $q=0$ a.e. (ii) The procedure in Section \ref{asyl} works for the anti-periodic boundary conditions \begin{equation*}\label{antiper} y(0)=-y(a),\quad y'(0)=-y'(a). \end{equation*} Thus, it is readily seen that the corresponding results for the anti-periodic eigenvalues $\mu_{2m}$, $\mu_{2m+1}$ hold, replacing $(2m+2)$ in \eqref{asy2}-\eqref{m1} by $(2m+1)$. \begin{thebibliography}{10} \bibitem{Ambarz} V.~Ambarzumian; \emph{\"{U}ber eine Frage der Eigenwerttheorie}, Zeitschrift f{\"{u}}r Physik 53 (1929) 690--695. \bibitem{Borg} G.~Borg; \emph{Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe bestimmung der differentialgleichung durch die eigenwerte}, Acta Math. 78 (1946) 1--96. \bibitem{anoteinver} Y.~H. Cheng, T.~E. Wang, C.~J. Wu; \emph{A note on eigenvalue asymptotics for Hill's equation}, Appl. Math. Lett. 23~(9) (2010) 1013--1015. \bibitem{corri} H.~H. Chern, C.~K. Lawb, H.~J. Wang; \emph{Corrigendum to \text{“Extension of Ambarzumyan's theorem} to general boundary conditions}, J. Math. Anal. Appl. 309 (2005) 764--768. \bibitem{Coddington:Levinson} E.~A. Coddington, N.~Levinson; \emph{Theory of Ordinary Differential Equations}, McGraw-Hill, New York, 1955. \bibitem{Coskun:invers} H.~Coskun; \emph{Some inverse results for Hill's equation}, J. Math. Anal. Appl. 276 (2002) 833--844. \bibitem{Eastham} M.~S.~P. Eastham; \emph{The Spectral Theory of Periodic Differential Operators}, Scottish Academic Press, Edinburgh, 1973. \bibitem{FreilingYurko} G.~Freiling, V.~A. Yurko; \emph{Inverse \text{Sturm–Liouville} Problems and Their Applications}, NOVA Science Publishers, New York, 2001. \bibitem{Goldberg:determination} W.~Goldberg; \emph{On the determination of a \text{Hill}'s equation from its spectrum}, J. Math. Anal. Appl. 51~(3) (1975) 705--723. \bibitem{Goldberg:necessary} W.~Goldberg; \emph{Necessary and sufficient conditions for determining a Hill's equation from its spectrum}, J. Math. Anal. Appl. 55 (1976) 549--554. \bibitem{Goldberghoch:selected} W.~Goldberg, H.~Hochstadt; \emph{On a Hill's equation with selected gaps in its spectrum}, J. Differential Equations 34 (1979) 167--178. \bibitem{Goldberghoch:finitenumber} W.~Goldberg, H.~Hochstadt; \emph{On a periodic boundary value problem with only a finite number of simple eigenvalues}, J. Math. Anal. Appl. 91 (1982) 340--351. \bibitem{Hochstadt:Stability-Estimate} H.~Hochstadt; \emph{Estimates on the stability intervals for the Hill's equation}, Proc. Amer. Math. Soc. 14 (1963) 930--932. \bibitem{Hochstadt:determination} H.~Hochstadt; \emph{On the determination of a \text{Hill}'s equation from its spectrum}, Arch. Rational Mech. Anal. 19 (1965) 353--362. \bibitem{Magnus-Winkler} W.~Magnus, S.~Winkler; \emph{Hill's Equations}, Interscience Publishers, Wiley, 1969. \bibitem{Marchenko} V.~A. Marchenko; \emph{Sturm-Liouville Operators and Applications}, vol.~22 of Oper. Theory Adv., Birkhauser, Basel, 1986. \bibitem{McKean} H.~McKean, E.~Trubowitz; \emph{Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points}, Comm. Pure Appl. Math. 29 (1976) 143--226. \bibitem{Poschel} J.~P{\"{o}}schel, E.~Trubowitz; \emph{Inverse Spectral Theory}, Academic Press, Boston, 1987. \bibitem{kyrac:titch} A.~A. K\i ra\c{c}; \emph{On the asymptotic simplicity of periodic eigenvalues and Titchmarsh's} formula, J. Math. Anal. Appl. 425~(1) (2015) 440 -- 450. \bibitem{kyrac:abstract} A.~A. \text{K\i ra\c{c}}; \emph{On the riesz basisness of systems composed of root functions of periodic boundary value problems}, Abstract and Applied Analysis 2015~(Article ID 945049) (2015) 7 pages. \bibitem{kırac:ambars} A.~A. \text{K\i ra\c{c}}; \emph{On the \text{Ambarzumyan's theorem} for the quasi-periodic problem}, Analysis and Mathematical Physics, http://dx.doi.org/10.1007/s13324-015-0118-0, (2015) 1--4. \bibitem{Trubowitz} E.~Trubowitz; \emph{The inverse problem for periodic potentials}, Comm. Pure Appl. Math. 30 (1977) 321--337. \bibitem{Ungar} P. Ungar; \emph{Stable \text{Hill} equations}, Comm. Pure Appl. Math. 14 (1961) 707--710. \bibitem{veliev;arşiv} O.~A. Veliev; \emph{Asymptotic analysis of non-self-adjoint Hill operators}, Central European Journal of Mathematics 11~(12) (2013) 2234--2256. \bibitem{Melda.O} O.~A. Veliev, M.~Duman; \emph{The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential}, J. Math. Anal. Appl. 265 (2002) 76--90. \bibitem{Veliev:Shkalikov} O.~A. Veliev, A.~A. Shkalikov; \emph{On the \text{Riesz} basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems}, Mathematical Notes 85~(5-6) (2009) 647--660. \bibitem{Ambarzcoupled} C.~F. Yang, Z.~Y. Huang, X.~P. Yang; \emph{Ambarzumyan's theorems for vectorial sturm-liouville systems with coupled boundary conditions}, Taiwanese J. Math. 14~(4) (2010) 1429--1437. \end{thebibliography} \end{document}