Friday, August 2, 2013

1305.2595 (Alexander Moroz)

On algebraic solvability of the Rabi model    [PDF]

Alexander Moroz
A novel method of solving Rabi-like models is presented. In the case of the Rabi model the method yields an analytic solution that is considerably simpler and the number of ca. {\em 1350} calculable energy levels per parity subspace obtained by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak's solution in double precision. The eigenfunctions can be determined in terms of polynomials $\phi_{k}(\upepsilon)$. Any first $n$ eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of $\phi_{N}(\upepsilon)$ of at least the degree $N=n+n_t$. The value of $n_t>0$, which is slowly increasing with $n$, depends on the required precision. For instance, $n_t\simeq 26$ for $n=1000$ and $\kappa=0.2$ if double precision is required. Given that the sequence of the $l$th zeros $x_{nl}$'s of $\phi_{n}(\upepsilon)$'s defines a monotonically decreasing discrete flow with increasing $n$, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for $\kappa\le 1$, numerics and explicit example suggest that the main conclusions remain to be valid also for $\kappa>1$.
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