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In this paper the inverse scattering problem is considered for a version of the one-dimensional Schrödinger equation with turning point on the half-line (0, ∞). The scattering data of the problem is defined and the fundamental equation is derived. With the help of the derived fundamental equation, in terms of the scattering data, the potential is recovered uniquely.MSC: 58C40; 34L25; 34B05; 47A40

In this paper we introduce and investigate the eigenvalues and the normalizing numbers as well as the scattering function for some version of the one-dimensional Schrödinger equation with turning point on the half line. MSC: 58C40; 34L25

We prove the eigenfunction expansion formula for a Dirichlet problem with explosive factor by two ways, first by standard method and second by proving a convergence in some metric space .