  In this paper we solve a problem about the Schr$\ddot{o}$dinger operator with
potential $v(\theta)=2\lambda cos2\pi\theta/(1-\alpha cos2\pi\theta),\
(|\alpha|<1)$ in physics. With the help of the formula of Lyapunov exponent in
the spectrum, the coexistence of zero Lyapunov exponent and positive Lyapunov
exponent for some parameters is first proved, and there exists a curve that
separates them. The spectrum in the region of positive Lyapunov exponent is
purely pure point spectrum with exponentially decaying eigenfunctions for
almost every frequency and almost every phase. From the research, we realize
that the infinite potential $v(\theta)=2\lambda tan^2(\pi\theta)$ has zero
Lyapunov exponent for some energies if $0<|\lambda|<1$.
