We introduce an exactly solvable one-dimensional potential that supports both bound and/or resonance states. This potential is a generalization of the well-known 1D Morse potential where we introduced a deformation that preserves the finite spectrum property. On the other hand, in the limit of zero deformation, the potential reduces to the exponentially confining potential well introduced recently by Alhaidari [Theor. Math. Phys. 206, 84–96 (2021)]. The latter potential supports an infinite spectrum, which means that the zero deformation limit is a critical point where our system will transition from the finite spectrum limit to the infinite spectrum limit. We solve the corresponding Schrodinger equation and obtain the energy spectrum and the eigenstates using the tridiagonal representation approach.
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In our work, we define a resonance state as a short-lived eigenstate of the Hamiltonian operator with an energy eigenvalue that has a positive real part that lies between the local maxima and the local minima of the potential. Therefore, a necessary, but not sufficient, condition for the existence of resonance is that the cubic equation (2) has two separate positive real roots in the interval 0 < z0 < 1/q.
