Quantum mechanical analysing of generalized woods-saxon potential in one dimension for scattering, bound and quasi-bound states

In this work, the exact analytical solutions of the one-dimensional Schr$\ddot{o}$dinger equation for the generalized symmetric Woods-Saxon potential are solved for the scattering and bound states. There is a various information about Woods-Saxon potential. The energy eigenvalues that gives the bound states are found. It is calculated analitically on the scattering states that protected probability density. Also, it is determined tunneling which conditions may occur depending parameters. The bound states for generalized Woods-Saxon potential are investigated as tight-bound state and quasi-bound state. Then, we obtained the energy eigenvalues spectrum for two states and determined wave functions with arbitrary parameters.