Krasovsky I.V.
A Matrix Commuting with the Square of the Almost Mathieu Operator.
(9K, LaTeX)

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
almost Mathieu" operator $H_\psi$
\begin{eqnarray}
(H_{\psi}\psi)_n=\psi_{n-1}+2\cos(\omega n+\theta)\psi_n+\psi_{n+1},\\
n=0,1,\dots,N-1; \psi_{-1}=\psi_{N-1}; \psi_{N}=\psi_{0};
\end{eqnarray}
in the case when $\omega=2\pi M/N$ (numbers $M$, $N$ are relatively prime
integers), $N$ is divisible by 4, and $\cos(N\theta)=-1$.