# Largest eigenvalues of two Harper matrices

Let $H_0, H_1$ be two Harper matrices, i.e. periodic Jacobi matrices of the form: $$J=(a_0, a_1, \ldots, a_{q-1}; 1,\ldots, 1; 1)$$ with diagonal entries $a_k=A\cos(2\pi k p/q+\theta)$, $k=\overline{0, q-1}$, $p,q$ relative prime integers, $q>0$, and $A\in(0,2)$.

For $J=H_0$, $\theta=0$, and for $J=H_1$, $\theta= 2\pi/(2q)$.

My question is: if $\lambda_0$ is the largest eigenvalue of $H_0$, and $\mu_0$ the largest eigenvalue of $H_1$ is there a result that relates these eigenvalues? More precisely, is it true that $\lambda_0>\mu_0$, for any $p,q$?

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