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In this work, we investigate the Anderson localization problems of thegeneralized Aubry-Andr\'{e} model (Ganeshan-Pixley-Das Sarma's model) with anunbounded quasi-periodic potential where the parameter $|\alpha|\geq1$. TheLyapunov exponent $\gamma(E)$ and the mobility edges $E_c$ are exactly obtainedfor the unbounded quasi-periodic potential. With the Lyapunov exponent, we findthat there exists a critical region in the parameter $\lambda-E$ plane. Thecritical region consists of critical states. In comparison with localized andextended states, the fluctuation of spatial extensions of the critical statesis much larger. The numerical results show that the scaling exponent of inverseparticipation ratio (IPR) of critical states $x\simeq0.5$. Furthermore, it isfound that the critical indices of localized length $\nu=1$ for bounded($|\alpha|<1$) case and $\nu=1/2$ for unbounded ($|\alpha|\geq1$) case. Theabove distinct critical indices can be used to distinguish thelocalized-extended from localized-critical transitions. At the end, we showthat the systems with different $E$ for both cases of $|\alpha|<1$ and$|\alpha|\geq1$ can be classified by the Lyapunov exponent $\gamma(E)$ andAvila's quantized acceleration $\omega(E)$.

Lyapunov exponent, mobility edges, and critical region in the generalized Aubry-Andre model with an unbounded quasiperiodic potential