We study global dynamics of solutions to the Cauchy problem for the focusing semilinear Schrödinger equation with a potential on the real line. The probleln is locally well-posed in the energy space. Our aim in this presentation is to study global behavior of the solution and prove a scattering result and a blow-up result for the problem with the data whose mass-energy is less than that of the ground state, where the ground state is the unique radial positive solution to the stationary Schrödinger equation without the potential. The scattering result for the defocusing version is recently studied by Lafontain.

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