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In this paper we solve a problem about the Schr$\ddot{o}$dinger operator withpotential $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 inthe spectrum, the coexistence of zero Lyapunov exponent and positive Lyapunovexponent for some parameters is first proved, and there exists a curve thatseparates them. The spectrum in the region of positive Lyapunov exponent ispurely pure point spectrum with exponentially decaying eigenfunctions foralmost every frequency and almost every phase. From the research, we realizethat the infinite potential $v(\theta)=2\lambda tan^2(\pi\theta)$ has zeroLyapunov exponent for some energies if $0<|\lambda|<1$.

Coexistence of zero Lyapunov exponent and positive Lyapunov exponent for new quasi-periodic Schr$\ddot{o}$dinger operator