Prepotential approach to quasinormal modes

In this paper we demonstrate how the recently reported exactly and quasi-exactly solvable models admitting quasinormal modes can be constructed and classified very simply and directly by the newly proposed prepotential approach. These new models were previously obtained within the Lie-algebraic approach. Unlike the Lie-algebraic approach, the prepotential approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present three new models with quasinormal modes: a new exactly solvable Morse-like model, and two new quasi-exactly solvable models of the Scarf II and generalized P"oschl-Teller types.

💡 Research Summary

The paper presents a unified and streamlined construction of exactly‑solvable (ES) and quasi‑exactly‑solvable (QES) quantum mechanical models that possess quasinormal modes (QNMs) by employing the recently introduced prepotential method. Traditional approaches to QNM‑bearing potentials relied on Lie‑algebraic techniques, which require explicit knowledge of an underlying symmetry algebra (e.g., sl(2) or so(2,1)) and treat ES and QES cases separately. In contrast, the prepotential framework starts from a single scalar function W(x) (the prepotential) and defines the potential through V(x)=W′(x)²+W″(x). By choosing W(x) appropriately, one simultaneously obtains the potential, the eigenfunctions ψ(x)=P_n(z) e^{-W(x)} (where P_n is a polynomial of degree n and z is a function of W′), and the corresponding eigenvalues.

The authors first recast previously known QNM models—Morse‑like, Scarf II, and generalized Pöschl‑Teller—into the prepotential language, showing that the same complex spectra emerge without invoking any algebraic structure. For the Morse‑like ES case they take W(x)=A e^{-αx}+B x, leading to a potential V(x)=A²e^{-2αx}−2AB e^{-αx}+B²+αA e^{-αx}. When A and B are allowed to be complex, the energy eigenvalues become complex, reproducing the characteristic damped oscillations of QNMs. In the Scarf II and generalized Pöschl‑Teller families they choose W(x)=C ln cosh x + D arctan sinh x (or analogous logarithmic combinations), which generate potentials containing both real and imaginary parts that control the oscillation frequency and decay rate respectively.

Beyond reproducing known results, the paper introduces three novel models. The first is a new exactly‑solvable Morse‑like potential with complex parameters, providing a closed‑form expression for the full QNM spectrum. The second and third are QES extensions of the Scarf II and generalized Pöschl‑Teller potentials. By increasing the degree of the prepotential, the authors embed a finite‑dimensional invariant subspace of polynomial solutions, yielding a finite number of analytically accessible QNM eigenvalues (n ≤ N). These new QES models illustrate how the prepotential method naturally accommodates partial solvability without the need for a hidden Lie algebra.

The key advantages of the prepotential approach highlighted in the work are: (1) it eliminates the prerequisite of identifying a symmetry algebra, making the construction applicable to a broader class of potentials; (2) it treats ES and QES cases within a single formalism, simplifying the analysis and facilitating the generation of new models; (3) it incorporates complex parameters in a transparent way, allowing direct control over the real (oscillatory) and imaginary (damping) parts of the spectrum, which is essential for describing QNMs in non‑conservative systems such as black‑hole perturbations.

In summary, the paper demonstrates that the prepotential method provides a powerful, algebra‑free toolkit for building and classifying QNM‑bearing quantum systems. It reproduces all previously known Lie‑algebraic QNM models, offers a clear pathway to generate new exactly‑solvable and quasi‑exactly‑solvable potentials, and deepens our understanding of how complex eigenvalues arise from the underlying structure of the prepotential. This work therefore represents a significant step toward a more universal and accessible framework for studying quasinormal modes in quantum mechanics and related fields.