Novel quasi-exactly solvable models with anharmonic singular potentials

We present new quasi-exactly solvable models with inverse quartic, sextic, octic and decatic power potentials, respectively. We solve these models exactly via the functional Bethe ansatz method. For each case, we give closed-form solutions for the energies and the wave functions as well as analytical expressions for the allowed potential parameters in terms of a set of algebraic equations.

💡 Research Summary

The paper introduces a family of quasi‑exactly solvable (QES) quantum‑mechanical models whose potentials contain singular inverse‑power terms of fourth, sixth, eighth and tenth order. The authors start from the three‑dimensional radial Schrödinger equation with a centrifugal barrier ℓ(ℓ+1)/r² and a potential of the form

V(r)=∑_{k∈{4,6,8,10}} a_k / r^{k} + b r²,

where a_k and b are real parameters. By a standard separation of variables the problem reduces to a one‑dimensional radial equation.

To obtain analytic solutions the wavefunction is assumed in the ansatz

ψ(r)=r^{ℓ} exp