Exact analytical spectra of an effective field-modified radial Schr\"odinger equation by the Nikiforov--Uvarov method

Abstract

We revisit the effective radial Schr\"odinger equation for a charged spinless particle in collinear static electric and magnetic fields that was formulated in our earlier work within the tridiagonal representation approach (TRA). In that model the angular factor of the electric coupling is replaced by the orientation parameter $C=\cos\theta$, the linear Zeeman term is diagonal in the azimuthal quantum number, and the effective field-modified central potential takes the form $V(r)=\Vt(r)+qECr$. Here we show that the same equation can be solved exactly without the TRA, by the Nikiforov--Uvarov (NU) method, once the potential component $\Vt(r)$ is prescribed. Three examples are worked out in closed form: a Kratzer-type component, which yields a linear-plus-Coulomb-plus-inverse-square central potential; a pseudoharmonic-type component, which yields a linear-plus-harmonic-plus-inverse-square potential; and a Hulth\'en-type component with a repulsive core (Manning--Rosen form), which yields a screened confining potential with a finite number of $s$-wave bound states. For each case we obtain the energy spectrum, the normalized eigenfunctions in terms of Laguerre or Jacobi polynomials, the location of the potential minimum, and closed-form radial moments. The spectra are independent of the orientation $\theta$ and of the electric field, in agreement with the structure found by the TRA, while the field enters through the shape of $V(r)$, the radial moments, and the expectation value $\langle V\rangle_k$. All analytical results are confirmed by an independent finite-difference diagonalization.