Quasi-exactly solvable models based on special functions

Abstract

We suggest a systematic method of extension of quasi-exactly solvable (QES) systems. We construct finite-dimensional subspaces on the basis of special functions (hypergeometric, Airy, Bessel ones) invariant with respect to the action of differential operators of the second order with polynomial coefficients. As a example of physical applications, we show that the known two-photon Rabi Hamiltonian becomes quasi-exactly solvable at certain values of parameters when it can be expressed in terms of corresponding QES operators related to the hypergeometric function.Comment: 15 pages. Typos correcte

Symmetric rotating-wave approximation for the generalized single-mode spin-boson system

“…We first note a symmetry of this system that is not present in H 1 , namely that H 2 does not couple even and odd Fock states. This has been discussed in a different context [32] and stems from the nature of the quadratic coupling in the squeezed harmonic oscillator H ± 2 from Eq. (8).…”

“…Additionally, counter-rotating terms dominate the short-time dynamical behavior for some parameter regions, leading to important Zeno and anti-Zeno effects that are not reproduced by the RWA [24][25][26].The Hamiltonian of a two-level system coupled quadratically to a quantum harmonic oscillator, the twophoton Rabi Hamiltonian, has also been studied within the RWA [27]. Limitations of the RWA have been outlined for this system [28], but so far limited effort has been directed to studying it outside of the RWA [29][30][31][32]. This Hamiltonian arose in quantum optics as a phenomenological model for a three-level system interacting with two photons [28,30,33] and is also relevant in modeling pure dephasing in crystals [34].…”

Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations

“…With the variable change x = √ 2w 1/4 and the gauge transformation w −9/8 H 2 w 5/8 , we find H 2 (2J+l+2, −J−1/2) becomes proportional up to an additive constant to the differential operator J + 5 presented in [17] for the cases when E = 0 and l is an integer. To reproduce the solutions (3.13), the invariant subspace given in [17] for J + 5 must be extended to include Bessel functions multiplied by certain rational powers of x. We leave further details of these exact wavefunctions and the investigation of the cases when l is not an integer to future work.…”

“…Differential operators that act invariantly on a subspace spanned by polynomials multiplied by special functions of either hypergeometric, Airy or Bessel type have been constructed in [17]. With the variable change x = √ 2w 1/4 and the gauge transformation w −9/8 H 2 w 5/8 , we find H 2 (2J+l+2, −J−1/2) becomes proportional up to an additive constant to the differential operator J + 5 presented in [17] for the cases when E = 0 and l is an integer.…”

The quantum Rabi model: solution and dynamics

“…The eigenspectrum of this model has been obtained and discussed via the G-function approach by various authors [41,117,118,120,121,122,123]. Some isolated exact solutions have also been obtained [64,119,146,147,148]. The eigenstate |ψ of the two-photon quantum Rabi hamiltonian H is again…”