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preliminary version
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sum rules, Jacobi matrices
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\begin{document}
\title[On a a spectral property of Jacobi matrices]{On a spectral property of Jacobi matrices}

 
\author{S. Kupin}
\email{kupin@math.brown.edu}

\thanks{{\it Keywords:} Jacobi matrices, sum rules.}

\date{October 5, 2002}

\address{Department of Mathematics, Box 1917, Brown University, Providence RI 02912, USA.}

\begin{abstract}
Let $J$ be a Jacobi matrix with elements $b_k$ on the main diagonal and elements $a_k$ on the auxiliary ones.  We suppose that $J$ is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of $J$ coincides with $[-2,2]$, and its discrete spectrum is a union of two sequences $\{x^\pm_j\}, x^+_j\ge2, x^-_j\le-2$, tending to $\pm2$. We denote sequences $\{a_{k+1}-a_k\}$ and $\{a_{k+1}+a_{k-1}-2a_k\}$ by $\pt a$ and $\pt^2 a$, respectively.

The main result of the note is the following theorem.

\smallskip\nt
{\bf Theorem.}\quad
Let $J$ be a Jacobi matrix described above and $\s$ be its spectral measure. Then
$a-1,b\in l^4,\  \pt^2 a,\pt^2 b \in l^2$, if and only if
$$
i)\ \int^2_{-2} \log \s'(x) (4-x^2)^{5/2}\, dx>-\infty,\qquad
ii)\ \sum_j(x^\pm_j\mp2)^{7/2}<\infty. 
$$
\end{abstract}

\maketitle

\vspace{-0.7cm}
\section*{Introduction}\label{s0}
Let us consider a Jacobi matrix
$$
J=J(a,b)=\lb
\begin{array}{ccc}
b_0&a_0&0\\
a_0&b_1&\dots\\
\vdots&\vdots&\ddots
\end{array}\rb,
$$
defined by two sequences $a=\{a_k\},\ a_k>0$, and $b=\{b_k\},\  b_k\in\br$.
We suppose that $J$ is a compact perturbation of the so-called free Jacobi matrix $J_0$,
$$
J_0=\lb
\begin{array}{ccc}
0&1& 0\\
1&0&\dots\\
\vdots&\vdots&\ddots
\end{array}\rb.
$$
The (scalar) spectral measure $\s$ of $J$ is defined by the relation
$$
((J-z)^{-1}e_0,e_0)=\int_\br\frac{d\s(x)}{x-z},
$$
with $z\in\bc\bsl\br$. It is well-known that in this case the continuous part $\s_c$ of the measure is supported by $[-2,2]$, and its discrete part $\s_d$ lives on the union of two sequences $\{x^\pm_j\}$ such that $x^+_j\ge2,\ x^-_j\le-2$,  and 
$x^\pm_j\to\pm 2$, as $j\to\infty$.

Let us put $\pt a=\{a_{k+1}-a_{k}\}$ and $\pt^2 a=\{a_{k+1}+a_{k-1}-2a_k\}$ and denote by $1$ the sequence consisting of ones. The following theorem holds.

\begin{theorem}\label{t1}
Let $J=J(a,b)$ and $\s$ be its spectral measure. Then
\begin{equation}\label{e1}
a-1,b\in l^4,\ \pt^2 a,\pt^2 b \in l^2,
\end{equation} 
if and only if
\begin{equation}
i)\ \int^2_{-2} \log \s'(x) (4-x^2)^{5/2}\, dx>-\infty,\quad
ii)\ \sum_j(x^\pm_j\mp2)^{7/2}<\infty.  \label{e2}
\end{equation}
\end{theorem}
The theorem has an immediate corollary.
\begin{corollary}\label{c1}
Let $J=J(a,b)$ and $\s$ be its spectral measure. If
$a-1,b\in l^4,\ \pt a,\pt b \in l^2$,
then relations \eqref{e2} hold.
 \end{corollary}
In this direction we may also prove the following proposition.
\begin{proposition}\label{p1}
Let $J=J(a,b)$ and $\s$ be its spectral measure. If
$a-1,b\in l^3,\ \pt a,\pt b \in l^2$, then
\begin{equation*}
i)\   \int^2_{-2} \log \s'(x) (4-x^2)^{3/2}\, dx>-\infty, \quad
ii)\  \sum_j(x^\pm_j\mp2)^{5/2}<\infty.   
\end{equation*}
\end{proposition}
Together with Corollary \ref{c1}, this proposition leads to a natural conjecture. The conjecture  would yield a right discrete counterpart of results, obtained in \cite{mnv} for Schr\"odinger operators on the half-line.
\begin{conjecture}
For an integer $k\ge2$,  conditions $a-1,b\in l^k,\ \pt a,\pt b \in l^2$ imply
\begin{equation*}
i)\  \int^2_{-2} \log \s'(x) (4-x^2)^{k-3/2}\, dx>-\infty,\quad
ii)\ \sum_j(x^\pm_j\mp2)^{k-1/2}<\infty.  
\end{equation*}
\end{conjecture}
The author would like to thank B. Simon and P. Yuditskii for helpful discussions on the subject.

\section{Proof of Theorem \ref{t1}}\label{s1}
It is convenient to map the domain $\hat\bc\bsl [-2,2]$ onto the unit disk with the help of transformation $\z(z)=1/2(z-\sqrt{z^2-4})$. 
We construct a measure $\mu$ on the unit circle letting $d\mu(\th)=d\s(x)/(2\sin\th)$, where $x=z(e^{i\th})$ and $z(\z)=\z+1/\z$. 
The sequences $\{x^\pm_j\}$ are mapped to points $\{\z_j\}$, lying on the real diameter of the unit disk. In these new terms Theorem \ref{t1} says that
$a-1,b\in l^4,\ \pt^2 a,\pt^2 b \in l^2$ if and only if
\begin{equation}\label{e3}
i)\  \int^{2\pi}_0\log\mu'(\th)\sin^6\th\, d\th>-\infty,\quad ii)\ \sum_j(1-|\z_j|)^7<\infty.
\end{equation}

Let us assume first that $\mathrm{rank}\, (J-J_0)<\infty$. For $j\ge1$, we define Chebyshev polynomials by recurrence relations
$T_{j+1}(z)=zT_j(z)-T_{j-1}(z)$, and we take $T_0(z)=2, T_1(z)=z$. We have the so-called sum rules \cite{c1,c2,ks,te}
\begin{eqnarray*}
0)&& \frac1{4\pi}\int^{2\pi}_0\log\frac{2\sin\th}{\mu'(\th)}\, d\th=\sum_j \log\b_j-\sum_j\log a_j,\\
n)&& \frac1{2\pi}\int^{2\pi}_0\log\frac{2\sin\th}{\mu'(\th)}\cos n\th\, d\th=\frac1n\sum_j (\b^n_j-\b^{-n}_j)-\frac1n\tr\{T_n(J)-T_n(J_0)\},
\end{eqnarray*}
where we let $\b_j=1/|\z_j|$.
Taking into account that $\sin^6\th=1/2^5(10-15\cos 2\th+6\cos 4\th-\cos 6\th)$, we get
$$
\frac1{2\pi}\int^{2\pi}_0\log\frac{2\sin\th}{\mu'(\th)}\sin^6\th\, d\th+\frac16\sum_j F(\b_j)=\frac16\tr\{G(J)-G(J_0)-120\log A\},
$$ 
where $A=\di\{a_k\}$, and
\begin{eqnarray}
F(\b)&=&\frac1{2^5}\lt((\b^6-\b^{-6})-9(\b^4-\b^{-4})+45(\b^2-\b^{-2})-120\log\b\rt), \nonumber\\
G(J)&=&\frac1{2^5}\lt(T_6(J)-9T_4(J)+45T_2(J)\rt)=\frac1{2^5}(J^6-15J^4+90J^2-110).\label{e31}
\end{eqnarray}
Notice that $F(\b)=C_0(\b-1)^7+O((\b-1)^8)$, where $\b\ge1$ and $C_0$ is a positive constant.
 
Let 
\begin{equation}
\pp(a,b)=\pp(J(a,b))=\tr\{G(J)-G(J_0)-120\log A\}.\label{e32}
\end{equation} 
The following lemma is a cornerstone of the proof of the theorem.
\begin{lemma1}\label{l1}  Let $a=\{a_k\},\ b=\{b_k\}$ and $a_k\not=1, b_k\not=0$, for finitely many indices only. Let also 
norms $||a-1||_\infty,\ ||b||_\infty $ be small enough.  Then
\begin{equation}\label{e4}
\pp(a,b)\asymp(||a-1||^4_4+||b||^4_4+||\pt^2a||^2_2+||\pt^2b||^2_2),
\end{equation}  
In particular, $\pp(a,b)\ge 0$ for  these $a$ and $b$.
\end{lemma1}
The norms $||.||_p$  refer to the standard $l^p$-space norms. The sign ``$\asymp$'' means a two-sided estimate with positive constants depending on $||a-1||_\infty,\ ||b||_\infty$.
The lemma will be proved in the next section. 

With the exception of this lemma, the proof of Theorem \ref{t1} follows a well-known scheme \cite{ks,lns}.  We give only a sketch of its proof.

\medskip\nt
{\it Proof of Theorem \ref{t1}.}\quad 
Since the essential spectrum of $J=J(a,b)$ is the interval $[-2,2]$, we have (see \cite{de}) that $a_k\to1$ and $b_k\to0$. Consequently, we assume without loss of generality that norms $||a-1||_\infty$ and $||b||_\infty$ are small enough to apply the Main Lemma.

First, we show that condition \eqref{e1} implies \eqref{e3}, which is equivalent to showing \eqref{e2}. Denote
\begin{equation*}
\hh(J)=\hh(\mu)=\hh_1(\mu)+\hh_2(\mu)=\frac1{2\pi}\int^{2\pi}_0\log\frac{2\sin\th}{\mu'(\th)}\sin^6\th\,d\th+\frac16\sum_jF
\lt(\frac1{|\z_j|}\rt),
 \end{equation*}
Let  $a_N=\{(a_N)_k\}$ and $a'_N=\{(a'_N)_k\}$, where
$$
(a_N)_k=\lt\{ 
\begin{array}{lcl}
a_k,&& k\le N,\\
1,&&  k\ge N+1,
\end{array}\rt. 
\quad
(a'_N)_k=\lt\{ 
\begin{array}{lcl}
1,&& k\le N,\\
a_k,&& k\ge N+1.
\end{array}\rt. 
$$
We also consider $b_N, b'_N$, constructed in the same way (of course, with 1's replaced by 0's).
Let $J_N=J(a_N,b_N)$.  Obviously, we have that  $a'_N-1, b'_N\to 0$ and $\pt^2a'_N, \pt^2b'_N\to0$ in corresponding norms. Since 
$$
\pp(a,b)=\sum_k\psi(a_k, a_{k+1},a_{k+2};b_k, b_{k+1}, b_{k+2})
$$ 
for a function $\psi$ (see Sect. \ref{s2}), we have that
$$
|\pp(J)-\pp(J_N)|\le\pp(a'_N,b'_N)\le C_1(||a'_N-1||^4_4+||b'_N||^4_4+||\pt^2a'_N||^2_2+||\pt^2b'_N||^2_2)\to 0,
$$
or $\pp(J_N)\to \pp(J)$. From the other hand, $(J_N-z(\z))^{-1}\to(J-z(\z))^{-1}$, for $\z$ inside the unit disk and, consequently, $\mu_N\to\mu$ weakly. Using \cite{ks}, Corollary 5.3 and Theorem 6.2, we get
$$
\hh_1(\mu)\le\liminf_N \hh_1(\mu_N)
$$ 
and 
$$
\lim_{N\to\infty} \hh_2(\mu_N)=\hh_2(\mu).
$$
Summing up, we obtain
$$
\hh(\mu)\le\limsup_N\hh(\mu_N)=\limsup_N\pp(J_N)=\lim_{N\to\infty} \pp(J_N)=\pp(J).
$$

We now show the converse, i.e. condition \eqref{e3} yields relations \eqref{e1}. Recall (\cite{ks}, Proposition 8.4) that if $\mu=\mu(J)$ satisfies inequality $\mu\ge\d m$, then $$
\pp(J)\le\hh(J),
$$ 
where $\d>0$ and $m$ is the Lebesgue measure on the unit circle. We take a $J$ with the property $\pp(J)<\infty$, and we put $\mu_\d=(1-\d)\mu+\d m$ for $\mu=\mu(J)$ and $\d\in(0,1)$. The measure $\mu_\d$ defines a Jacobi matrix $J_\d=J(a_\d,b_\d)$. Theorem 8.1 of \cite{ks}  shows that $\hh(J)=\lim_{\d\to 0}\hh(J_\d)$. On the other hand (see \cite{ks}, Proposition 8.4 and Theorem 8.1), 
$$
\pp(J)\le\liminf_\d\pp(J_\d).
$$
Hence,
$$
0\le\pp(J)\le\liminf_\d \pp(J_\d)=\liminf_\d \hh(J_\d)=\lim_{\d\to 0} \hh(J_\d)=\hh(J).
$$ 
The theorem is proved. \hfill$\Box$

\section{Proof of the main lemma}\label{s2}
We make few simplifications before going into the proof. First, we drop the term $\tr G(J_0)$ in \eqref{e32}, since it is finite. Second, we discard factors $1/2^5$ (see \eqref{e31}).
Having two sequences $a=\{a_k\}$ and $b=\{b_k\}$, we denote the sequence $\{a_kb_k\}$ by $ab$. We also write $a^\pm$ 
for the sequence $\{a_{k\pm 1}\}$.  

The starting point of the proof is computation of $\tr J^n$.  According to formulas from \cite{te}, Section 6.1, we have $\tr J^j=\sum_k(g_j)_k$, where sequences $g_j$ are constructed as
\begin{eqnarray}
g_{j+1}&=&h_j+h^-_j+bg_j, \label{e5}\\
h_{j+1}&=&a^2\sum^j_{l=0}g_{j-l}g^-_l-\sum^j_{l=0} h_{j-l}h_l,\nonumber
\end{eqnarray}
with $g_0=1, h_0=0$.

We give the proof of the lemma in cases when $J=J(a,b)$ with $b=0$, and $J=J(a,b)$ with $a=1$. Since the  proof in the general case follows  along the same lines and is  long and tedious, we omit it.

Let us consider the case when $b=0$.

\begin{lemma}\label{l2} Let $a=\{a_k\}$ and $a_k\not =1$ for only finitely many indices. Then
$$
\begin{array}{rcl}
\pp(a)=\pp(J(a,0))&=&\sum_k \{-9\l_k+3\l^2_k\l_{k+1}+3\l\l^2_{k+1}+3\l_{k-1}\l_k \l_{k+1}\\
&&\\
&+&9\l^2_k-12\l\l_{k+1}+3\l_{k-1}\l_{k+1}+\frac{15}2 \l_k^4+O(|\l_k|^5)\},
\end{array}
$$
where $\l=a^2-1$.
\end{lemma}

\begin{proof} A straightforward computation using \eqref{e5} yields $g_1=g_3=g_5=0$ and
$$
\begin{array}{rcl}
g_2&=&a^2+a^{-2},\\
g_4&=&a^{-2}(a^{--2}+a^{-2}+a^2)+a^2(a^{-2}+a^2+a^{+2}),\\
g_6&=&a^2(a^{-2}(a^{--2}+\dots+a^{+2})+a^{2}(a^{-2}+\dots+a^{+2})+a^{+2}(a^{2}+\dots+a^{++2}))\\
      &+&a^{-2}(a^{--2}(a^{---2}+\dots+a^{2})+a^{-2}(a^{--2}+\dots+a^{2})+a^{2}(a^{-2}+\dots+a^{+2})).
\end{array}
$$
We also have $\tr\log A=1/2\sum_k\log(1+\l_k)$. Rewriting in terms of $\l=a^2-1$ and recalling that $\tr J^j=\sum_k(g_j)_k$, we get the conclusion of the lemma. 
\end{proof}

In particular, we see that
$$
\pp(a)=\sum_k \lt\{\psi_1(\l_k,\l_{k+1},\l_{k+2})+O(|\l_k|^5)\rt\},
$$
where
$$
\begin{array}{rcl}
\psi_1(x,y,z)&=&\dsp\frac12(x^3-20y^3+z^3+6xy^2+6y^2z+6xyz)\\
&&\\
&+&\dsp\frac32(x^2+4y^2+z^2-4xy-4yz+2xz)+\frac52(x^4+4y^4+z^4).
\end{array}
$$
Note also that $\l_k=2(a_k-1)+O((a_k-1)^2)$.

\begin{lemma}\label{l3}
We have
$$
\psi_1(x,y,z)\asymp (x^4+y^4+z^4+(x+z-2y)^2).
$$
\end{lemma}

\begin{proof}
We notice that
\begin{eqnarray}
\psi_1(x,y,z)&=&\frac12((x+z-2y)(x^2+10y^2+z^2+2xy+2yz-zx)  \label{e6}\\
&+&3(x+z-2y)^2)+5(x^4+4y^4+z^4)), \nonumber
\end{eqnarray}
and  the bound from above follows from the inequality $ab\le1/2(a^2+b^2)$ for $a,b\ge0$. To get the bound from below, we estimate the first term in \eqref{e6} as
$$
|(x+z-2y)(x^2+10y^2+z^2+2xy+2yz-zx)|\le \frac12(x+z-2y)^2+\frac32(x^4+4y^4+z^4),
$$
and, consequently,
$$
\psi_1(x,y,z)\ge\frac12\lt(\frac52(x+z-2y)^2+\frac72(x^4+4y^4+z^4)\rt).
$$
The lemma is proved.
\end{proof}

Now, we turn to the second case $J=J(a,b)$ with $a=1$.

\begin{lemma}\label{l4}
Let $b=\{b_k\}$ and $b_k\not =0$ for only finitely many indices. Then
$$
\pp(b)=\pp(1,b)=\sum_k \lt\{b^6+(-3b^4+6bb^{+3}+6b^+b^3+6b^2b^{+2})+(18b^2-24bb^++6bb^{++})\rt\}.
$$
\end{lemma}
\begin{proof}
The argument is exactly as in Lemma \ref{l2}.  We only quote expressions for $g_5$ and $g_6$ 
$$
\begin{array}{rcl}
g_5&=&b^5+(b^{-3}+8b^3+b^{+3}+2b^{-2}b+3b^-b^2+3b^2b^++2bb^{+2})\\
&+&(b^{--}+6b^-+16b+6b^++b^{++}),\\
&&\\
g_6&=&b^6+(b^{-4}+10b^4+b^{+4}\\
&+&4b^-b^3+3b^{-2}b^2+2bb^{-3}+2bb^{+3}+3b^2b^{+2}+4b^3b^+)\\
&+&(b^{--2}+8b^{-2}+30b^2+8b^{+2}+b^{++2}+2b^-b^{--}+2bb^{--}\\
&+&16bb^-+16bb^++2bb^{++}+2b^+b^{++}+2b^+b^-)+20.
\end{array}
$$
\end{proof}
Hence, we obtain 
$$
\pp(b)=\sum_k\{\psi_2(b_k,b_{k+1},b_{k+2})+O(b^6_k)\},
$$
where
$$
\psi_2(x,y,z)=-3y^4+(6xy^3+6y^3z+3y^2x^2+3y^2z^2)+3(x+z-2y)^2.
$$
\begin{lemma}\label{l5}
We have
$$
\psi_2(x,y,z)\asymp (y^4+(x+z-2y)^2).
$$
\end{lemma}
\begin{proof}
One more time, the bound from above follows at once from inequality $ab\le1/p\,a^p+1/q\,b^q$, where $a,b>0$, and $1/p+1/q=1$. 

Furthermore, we see that
$$
\psi_2(x,y,z)=-3y^4+y^2\lt(\frac32(x-z)^2+3(x+y+z)^2-\frac12(x+z-2y)^2\rt)+3(x+z-2y)^2.
$$
Estimating moduli of negative terms like in Lemma \ref{l3} and taking into account
$$
-3y^4+3y^2(x+y+z)^2-\frac12y^2(x+z-2y)^2+3(x+z-2y)^2\ge \frac12(y^4+(x+z-2y)^2),
$$
finishes the proof.
\end{proof}
 

\begin{thebibliography}{9}

\bibitem{c1} K. Case, {\it Orthogonal polynomials from the viewpoint of scattering theory},   J. Mathematical Phys., {\bf 15}  (1974), 2166--2174.

\bibitem{c2} K. Case, {\it Orthogonal polynomials, II},  J. Mathematical Phys.,  {\bf 16}  (1975), 1435--1440.

\bibitem{de} S. Denisov,  {\it On the Nevai's conjecture and Rakhmanov's theorem for Jacobi matrices}, preprint.

\bibitem{ks} R. Killip, B. Simon, {\it Sum rules for Jacobi matrices and their applications to spectral theory}, to appear in Annals of Math.

\bibitem{lns} A. Laptev, S. Naboko, O. Safronov, {\it On new relations between spectral properties of Jacobi matrices and their coefficients},  to appear.

\bibitem{mnv} S. Molchanov, M. Novitskii, B. Vainberg, {\it  First KdV integrals and absolutely continuous spectrum for 1-D Schr\"odinger operator},
 Comm. Math. Phys.,  {\bf 216}  (2001),  no. 1, 195--213.

\bibitem{te} G. Teschl, {\it Jacobi operators and completely integrable nonlinear lattices},  AMS Mathematical Surveys and Monographs, 72, Providence, 2000. 

\end{thebibliography}

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