A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

Abstract

General concept of a gradation slicing is used to analyze polynomial solutions of ordinary differential equations (ODE) with polynomial coefficients, Lψ = 0, where L = l p l (z)d l z , p l (z) are polynomials, z is a one-dimensional coordinate, and d z = d/dz. It is not required that ODE is ei-Fuchsian or (ii) leads to a usual Sturm-Liouville eigenvalue problem. General necessary and sufficient conditions for the existence of a polynomial solution are formulated involving constraint relations. The necessary condition for a polynomial solution of nth degree to exist forces energy to a nth baseline. Once the constraint relations on the nth baseline can be solved, a polynomial solution is in principle possible even in the absence of any underlying algebraic structure. The usefulness of theory is demonstrated on the examples of various Rabi models. For those models, a baseline is known as a Juddian baseline (e.g. in the case of the Rabi model the curve described by the nth energy level of a displaced harmonic oscillator with varying coupling g). The corresponding constraint relations are shown to (i) reproduce known constraint polynomials for the usual anddriven Rabi models and (ii) generate hitherto unknown constraint polynomials for the two-mode, two-photon, and generalized Rabi models, implying that the eigenvalues of corresponding polynomial eigenfunctions can be determined algebraically. Interestingly, the ODE of the above Rabi models are shown to be characterized, at least for some parameter range, by the same unique set of grading parameters.

Constraint polynomial approach: an alternative to the functional Bethe Ansatz method?

“…Whereas in the Heun operator remains unchanged. Contrary to that, for a number of generalizations of the Rabi model [11] variations of spectral parameter induce translation on the corresponding baseline. This has the effect that to different values of spectral parameter there correspond different algebraic Heun operators (cf.…”

“…All those QES potentials lend themselves to sl 2 algebraization [6,7]. At the same time the above QES examples are used to elucidate essential difference with various generalizations of the Rabi model [11]. Whereas in the Heun operator remains unchanged.…”

“…Eq. (2) is a particular type of more general ordinary differential equation (ODE) with polynomial coefficients for which a general concept of gradation slicing has been recently employed in order to analyze their polynomial solutions [11]. Gradation slicing is universal and easily applicable algorithmic recursive approach for obtaining polynomial solutions which does not require any a priori knowledge about hidden algebraic structure of ODE.…”

“…Gradation slicing is universal and easily applicable algorithmic recursive approach for obtaining polynomial solutions which does not require any a priori knowledge about hidden algebraic structure of ODE. Its usefulness has been so far demonstrated on the examples of various Rabi models [11].…”

“…In the present article we employ the gradation slicing approach of Ref. [11] to determine polynomial solutions of quasi-exactly solvable (QES) Schrödinger equation for Xie [12] and Chen et al [13] three parameters modified Manning potentials [1,3,14], an electron in Coulomb and magnetic fields and relative motion of two electrons in an external oscillator potential [15,16], the perturbed double sinh-Gordon system (DSHG) [1,3,17], and the hyperbolic Razavy potential [1,3,18]. All those QES potentials lend themselves to sl 2 algebraization [6,7].…”