Quasi-exactly solvable Fokker-Planck equations

We consider exact and quasi-exact solvability of the one-dimensional Fokker-Planck equation based on the connection between the Fokker-Planck equation and the Schr"odinger equation. A unified consideration of these two types of solvability is given from the viewpoint of prepotential together with Bethe ansatz equations. Quasi-exactly solvable Fokker-Planck equations related to the $sl(2)$-based systems in Turbiner’s classification are listed. We also present one $sl(2)$-based example which is not listed in Turbiner’s scheme.

💡 Research Summary

The paper investigates exact and quasi‑exact solvability (QES) of the one‑dimensional Fokker‑Planck (FP) equation by exploiting its well‑known correspondence with the Schrödinger equation. By introducing a prepotential (W(x)) such that the drift term is (D^{(1)}(x)=2W’(x)) and the diffusion coefficient is constant ((D^{(2)}=2)), the FP operator can be mapped to a non‑Hermitian Schrödinger operator (H=-\partial_x^2+V(x)) with the effective potential (V(x)=W’^2-W’’). This mapping allows any solvable quantum‑mechanical model to be translated into a solvable FP model, provided the resulting probability density (P(x)=e^{-2W(x)}) remains normalizable.

The authors adopt the algebraic framework based on the Lie algebra (sl(2)). In this approach the Hamiltonian is expressed as a quadratic combination of the (sl(2)) generators (J^{\pm,0}). When the Hamiltonian preserves a finite‑dimensional polynomial space (\mathcal{V}_N=\text{span}{1,z,\dots ,z^{N-1}}) (with a suitable change of variable (z=z(x))), a finite number of eigenstates can be obtained exactly – the hallmark of QES. The coefficients of the polynomial eigenfunctions satisfy Bethe‑Ansatz equations (BAE), which are nonlinear algebraic conditions for the roots (z_i). Real, distinct roots guarantee a positive, normalizable probability density.

The paper first recasts the classic exactly solvable (ES) quantum models (harmonic oscillator, Morse, Pöschl‑Teller, Scarf, Rosen‑Morse, etc.) into FP form, presenting the corresponding prepotentials and BAE. It then systematically treats the nine (sl(2))‑based QES classes identified by Turbiner. Eight of these classes survive the additional physical constraints of the FP equation (positivity of diffusion, normalizability of the stationary distribution). For each class the authors list:

1. The explicit prepotential (W(x)) (often a rational, trigonometric, hyperbolic, exponential, or logarithmic function);
2. The associated Schrödinger‑type potential (V(x));
3. The change of variable (z(x));
4. The Bethe‑Ansatz equations governing the polynomial roots;
5. The admissible parameter ranges that ensure a well‑behaved stationary distribution.

A key contribution is the discovery of a new (sl(2))‑based QES FP model that does not appear in Turbiner’s original classification. The prepotential is chosen as \