%A Matrix Commuting with the Square of the Almost Mathieu Operator.
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\begin{document}
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\hfill {\small December 1996}
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\begin{center}
{\bf A Matrix Commuting with the Square of the Almost Mathieu Operator}\\
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I.V.Krasovsky\\
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B.I.Verkin Institute for Low Temperature Physics and Engineering\\
47 Lenina Ave., Kharkov 310164, Ukraine.\\
E-mail: ivk@igorvk.kharkov.ua
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\noindent
{\bf Abstract.} Using a transformation that reduces the almost Mathieu
operator to a tridiagonal form with zero main diagonal, a unitary matrix is
constructed that commutes with the square of the
$N$-dimensional operator $H_\psi$
\begin{eqnarray}
(H_{\psi}\psi)_n=\psi_{n-1}+2\cos(\omega n+\theta)\psi_n+\psi_{n+1},\nonumber\\
n=0,1,\dots,N-1;\quad \psi_{-1}=\psi_{N-1};\quad \psi_{N}=\psi_{0};\nonumber
\end{eqnarray}
in the case when $\omega=2\pi M/N$ (numbers $M$, $N$ are relatively prime
integers), $N$ is divisible by 4, and $\cos(N\theta)=-1$.
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As is well known, the problem of an electron on a two-dimensional lattice
subject to a perpendicular uniform magnetic field reduces to
consideration of Harper's equation (see, e.g., [\ref{Hof}--\ref{Kreft}])
in a Hilbert space $l^2(Z)$:
\begin{equation}
\eqalign{
(H\psi)_n=\varepsilon\psi_n;\\
(H\psi)_n=\psi_{n-1}+2\cos(\omega n+\theta)\psi_n+\psi_{n+1},\\
n=\dots,-1,0,1,\dots,\qquad \omega,\theta\in{\bf R}.}\label{1}
\end{equation}
$H$ is a particular case of the almost Mathieu operator. This operator
has been studied by many workers (see [\ref{Last1}] for a review).
When $\omega=2\pi M/N$, with relatively prime integers $M$, $N$
(that is they do
not have a common divisor other than 1), $H$ is an $N$-periodic tridiagonal
matrix and,
hence, its spectrum consists of $N$ intervals. When $\omega/2\pi$ is
irrational, the spectrum is a zero measure Cantor set. The latter fact
is proved for Lebesgue a.e. $\omega$ [\ref{Last2}]. For irrational
$\omega/2\pi$ the spectrum is independent of the parameter $\theta$.
Consider the $N$-dimensional operator $H_\psi$
defined as follows:
Let $\omega=2\pi M/N$, where $M$ and $N$ are relatively prime, and
write the equations (\ref{1}) for $n=0,1,\dots,N-1$ with periodic boundary
conditions, that is
\begin{equation}
\eqalign{
\psi_{n-1}+(q^ne^{i\theta}+q^{-n}e^{-i\theta})\psi_n+\psi_{n+1}=
\varepsilon\psi_n,\\
n=0,1,\dots,N-1;\qquad q=e^{i\omega};\qquad
\psi_{-1}=\psi_{N-1};\quad\psi_{N}=\psi_{0}.}\label{2}
\end{equation}
The corresponding $N\times N$ matrix I shall denote $H_\psi$, where
index $\psi$ reminds of the basis in which $H_\psi$ is defined. Later on,
similar notation for the same operator in other bases will be used.
Note, by the way,
that the discriminant $S_N(\varepsilon)$ (see, e.g., [\ref{Toda},\ref{prepr}])
associated with the $N$-periodic tridiagonal matrix
$H$ (\ref{1}) is equal to $(-1)^N\det(H_\psi-\varepsilon I)+2$,
where $I$ is the unity $N\times N$ matrix.
As $N\to\infty$, the almost Mathieu operator with
irrational $\omega/2\pi$ is obtained from $H_\psi$. In [\ref{WZprl},\ref{WZ}],
it was shown that the spectrum of matrices equivalent to $H_\psi$ for some
values of $\theta$ can be
represented as a solution of Bethe-ansatz-type algebraic equations.
In [\ref{FK}], a set of equations for arbitrary $\theta$ was derived.
In the present paper, a
certain operator $W$ is indicated that commutes with $H^2_\psi$ at
$\cos(N\theta)=-1$. Thus the spectral problem for $H_\psi$ is closely related
to the spectral problem for $W$. Let us now turn to exact constructions.
Substituting $\psi_n=\sum_{k=0}^{N-1}U_{nk}e^{ik\theta}\phi_k$, where
$U_{nk}=q^{nk}$, $n,k=0,1,\dots,N-1$, is, obviously, a unitary\footnote{
Henceforth unitary means unitary up to a factor, that is
$UU^*={\rm Const}\,I$.}
matrix, in (\ref{2}) and using the linear independence of the
columns of the transformation operator, we have
\begin{equation}
\eqalign{
\phi_{k-1}+(q^k+q^{-k})\phi_k+\phi_{k+1}=
\varepsilon\phi_k,\\
k=0,1,\dots,N-1;\qquad \phi_{-1}=e^{iN\theta}\phi_{N-1};\quad
\phi_{N}=e^{-iN\theta}\phi_{0}.}\label{3}
\end{equation}
In particular, if $\theta=0\,(\mod 2\pi)$ then we see that the commutator
$[H_\psi,U]=0$. This fact is not very valuable since $U^4=N^2I$ and,
consequently, $U/\sqrt{N}$ may have only 4 different eigenvalues: $i,-i,1,-1$.
Further, we shall consider the case when $N$ is even and
$N\theta=\pi\,(\mod 2\pi)$. These conditions ensure that after application
of the unitary transformation
\begin{equation}
\phi_k=\sum_{n=0}^{N-1}q^{\frac{1}{2}(k-1/2)^2+kn} \zeta_n\label{4}
\end{equation}
to (\ref{3}), we get a system with periodic boundary conditions:
\begin{equation}
\eqalign{
(1+q^{n-1})\zeta_{n-1}+(1+q^{-n})\zeta_{n+1}=
\varepsilon\zeta_n,\\
n=0,1,\dots,N-1;\qquad \zeta_{-1}=\zeta_{N-1};\quad
\zeta_{N}=\zeta_{0}.}\label{5}
\end{equation}
Due to the fact that (\ref{5}) does not contain
(unlike (\ref{2}) and (\ref{3})) the middle term
with $\zeta_n$, the square $H^2_\zeta$ splits into the direct sum of two
matrices: one is $H^2_{\zeta,1}$ defined in the basis
$\{e_{2n}\}_{n=0,1,\dots,N/2-1}$, the other is $H^2_{\zeta,2}$ in
$\{e_{2n+1}\}_{n=0,1,\dots,N/2-1}$,
where $\{e_n\}_{n=0,1,\dots,N-1}$ is the basis in which $H_\zeta$
(or $(H_\zeta-\varepsilon I)\zeta$, i.e. (\ref{5})) is written.\footnote
{Similar to (\ref{5}) representation without the middle term was first
obtained in [\ref{Kohmoto}] for the Hamiltonian of an electron on
a 2-dimensional lattice in magnetic field by changing the gauge of the
field. Such a change, roughly speaking, involves (unlike (\ref{4}))
not only $H$ but rather a direct sum of operators $H$ over $\theta$.}
Consider $H^2_{\zeta,1}$ first. We shall further assume that $N/4$ is
integer. Let us apply to $H^2_{\zeta,1}$ the unitary transformation
$\zeta_n=\sum_{k=0}^{N/2-1}q^{n^2+2nk}\tilde\xi_k$ and then shift the
basis by $N/4$: $\tilde\xi_{k+N/4}=\xi_k$. The purpose of the
shift is to get a matrix with zero boundary conditions:
\begin{equation}
\eqalign{
(1+q^{-1}-q^{-2k}-q^{2k-1})\xi_{k-1}+
(4-q^{2k}-q^{-2k}-q^{2k+1}-q^{-2k-1})\xi_k+\\
(1+q-q^{-2k-1}-q^{2k+2})\xi_{k+1}=
\varepsilon\xi_k,\qquad k=0,1,\dots,N/2-1;}\label{6}
\end{equation}
(We may assume any $\xi_{-1}$ and $\xi_{N/2}$.)
Now it is straightforward to verify that application of the transformation
$\xi_k=\sum_{p=0}^{N/2-1}\widetilde W_{kp}\varphi_p$,
$\widetilde W_{kp}=q^{(2k+1)p+(k+p)N/2}$, $k,p=0,1,\dots,N/2-1$ to (\ref{6})
leaves these equations unchanged. Thus $[H^2_{\xi,1},\widetilde W]=0$.
(Cf. $[H_{\psi},U]=0$.) The diagonal transformation
$V_{pm}=q^{pN/2}\delta_{pm}$ reduces $\widetilde W$ to a simpler form
\begin{equation}
W_{kp}=q^{(2k+1)p},\qquad k,p=0,1,\dots,N/2-1.
\end{equation}
Similarly, we find the basis in which $[H^2_{\gamma,2}, W']=0$,
where
\begin{equation}
{W'}_{kp}=q^{(2k+3)p},\qquad k,p=0,1,\dots,N/2-1.
\end{equation}
Unlike $U$, the unitary matrices $W$ and $W'$ have many different eigenvalues.
There exist corresponding eigenvectors
(for simple eigenvalues they are uniquely defined up to a factor) that,
as follows from the commutativity, are eigenvectors of $H^2$.
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{\it Phys.Rev.Lett.} {\bf 72}, 1890 (1994).
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\end{document}