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- W2780091465 abstract "The (driven) Rabi model, together with its two-mode, two-photon, and asymmetric generalizations, are exotic examples of quasi-exactly solvable models in that a corresponding 2nd order ordinary differential equation (ODE) ${cal L}psi=0$ with polynomial coefficients (i) is not Fuchsian one and (ii) the differential operator ${cal L}$ comprises energy E dependent terms $sim Ez d_z$, $Ez$, $E^2$. When recast into a Schrodinger equation (SE) form with the first derivative term being eliminated and the coefficient of $d_x^2$ set to one, such an equation is characterized by a nontrivially energy dependent potential. The concept of a gradation slicing is introduced to analyze polynomial solutions of such equations. It is shown that the ODE of all the above Rabi models are characterized by the same unique set of grading parameters. General necessary and sufficient conditions for the existence of a polynomial solution are formulated. Unlike standard eigenvalue problems, the condition that ${cal L}$ is an element of the enveloping algebra $mathcal U(sl_2)$ does not guarantee existence of a polynomial solution, because any such a solution has to be a zero mode of ${cal L}$ in an invariant $sl_2$ module. When one considers a coupling strength g of the Rabi model as a free parameter, the condition ${cal L}inmathcal U(sl_2)$ forces energy to a nth baseline, $nge 0$. However, on the nth continuous baseline there could be at most only n different isolated points where the Rabi model has a polynomial solution and any point on the baseline corresponds to a different ${cal L}inmathcal U(sl_2)$. The condition for ${cal L}inmathcal U(sl_2)$ is shown to merely coincide with the necessary condition for the existence of a polynomial solution of ${cal L}psi=0$. The gradation slicing concept enables one an efficient and convenient book keeping also in the Frobenius analysis." @default.
- W2780091465 created "2018-01-05" @default.
- W2780091465 creator A5061988286 @default.
- W2780091465 date "2017-12-26" @default.
- W2780091465 modified "2023-09-27" @default.
- W2780091465 title "A unified treatment of polynomial sectors of the Rabi models" @default.
- W2780091465 hasPublicationYear "2017" @default.
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