Closed-form solutions for the Schrödinger wave equation with non-solvable potentials: A perturbation approach

To obtain closed-form solutions for the radial Schrödinger wave equation with non-solvable potential models, we use a simple, easy, and fast perturbation technique within the framework of the asymptotic iteration method (PAIM). We will show how the PAIM can be applied directly to find the analytical coefficients in the perturbation series, without using the base eigenfunctions of the unperturbed problem. As an example, the vector Coulomb ( ∼ 1 / r ) \left( \sim 1\hspace{0.1em}\text{/}\hspace{0.1em}r) and the harmonic oscillator ( ∼ r 2 ) \left( \sim {r}^{2}) plus linear ( ∼ r ) \left( \sim r) scalar potential parts implemented with their counterpart spin-dependent terms are chosen to investigate the meson sectors including charm and beauty quarks. This approach is applicable in the same form to both the ground state and the excited bound states and can be easily applied to other strongly non-solvable potential problems. The procedure of this method and its results will provide a valuable hint for investigating tetraquark configuration.