Ucieczka z poetyki

Krajewska, Anna

doi:10.14746/pt.2016.26.0

Cited by 1 publication

(3 citation statements)

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“…Although the very notion of quantum integrability is the subject of ongoing dispute [31], it is largely accepted that if eigenvalues can be determined algebraically [11][12][13][14], this implies integrability and solvability [31]. Our results indicate that the algebraic solvability could be intricately linked with available precision.…”

Section: Algebraic Solvabilitymentioning

confidence: 72%

“…The very fact that the spectrum of a model is determined as zeros of a polynomial implies a special case of analytic solvability known as algebraic solvability [11][12][13][14][15]. Our another question is therefore Q2: Are there some models of the class R which are algebraically solvable?…”

Section: Hϕ = Eϕmentioning

confidence: 99%

“…The Rabi model is a typical example of quasi-exactly solvable (QES) models in quantum mechanics [11][12][13][14][15]. The QES models are distinguished by the fact that, for a chosen set of model parameters, a finite number of their eigenvalues and corresponding eigenfunctions can be determined algebraically [11][12][13][14][15]. In the case of the Rabi model [6], the latter eigenvalues correspond to the Juddian exact isolated solutions [28,29].…”

Section: Algebraic Solvabilitymentioning

confidence: 99%

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A hidden analytic structure of the Rabi model

Moroz

2014

Annals of Physics

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The Rabi model describes the simplest interaction between a cavity mode with a frequency $\omega_c$ and a two-level system with a resonance frequency $\omega_0$. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to $\Delta=\omega_0/(2\omega_c)=0$, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy $\upepsilon$, which are orthogonal on an equidistant lattice. A non-zero value of $\Delta$ leads to non-classical discrete orthogonal polynomials $\phi_{k}(\upepsilon)$ and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. {\em 1350} calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak's solution. 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 dimensionless interaction constant $\kappa=0.2$, if double precision is required. Although we can rigorously prove our results only for dimensionless interaction constant $\kappa< 1$, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for $\kappa\ge 1$.Comment: 10 pages, 3 figures - the amended versions gives more emphasis on the role played by discrete orthogonal polynomials in solving the Rabi model. New subsection IV.E summarizes open problems required to generalize the classical discrete Charlier polynomials describing the displaced harmonic oscillator into non-classical discrete polynomials describing the full Rabi mode

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