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Rectangular Potentials in a Semi-Harmonic Background: Spectrum, Resonances and Dwell Time

We study the energy properties of a particle in one dimensional semi-harmonic rectangular wells and barriers. The integration of energies is obtained by solving a simple transcendental equation. Scattering states are shown to include cases in which the impinging particle is 'captured' by the semi-harmonic rectangular potentials. The 'time of capture' is connected with the dwell time of the scattered wave. Using the particle absorption method, it is shown that the dwell time $τ^a_D$ coincides with the phase time $τ_W$ of Eisenbud and Wigner, calculated as the energy derivative of the reflected wave phase shift. Analytical expressions are derived for the phase time $τ_W$ of the semi-harmonic delta well and barrier potentials.