General Solution of the Schrödinger Equation for Some Hyperbolic Potentials

In this study, we obtain the recursive general solution of the Schrodinger equation $$y_{\nu }''(x;\lambda )+[\lambda -\nu (\nu +1)v(x)]y_{\nu }(x;\lambda )=0$$ for some Poschl–Teller type potentials when $$\nu =0,1,2,\ldots $$ . As a by product of the general solution, the finitely many bound states of the squared hyperbolic secant and tangent potentials are also derived when equipped with some suitable boundary conditions over the real line.