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Journal articles on the topic 'Quantum recurrence'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Kuznetsov, Vladimir. "Shock-wave model of the earthquake and Poincaré quantum theorem give an insight into the aftershock physics." E3S Web of Conferences 62 (2018): 03006. http://dx.doi.org/10.1051/e3sconf/20186203006.

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A fundamentally new model of aftershocks evident from the shock-wave model of the earthquake and Poincaré Recurrence Theorem [H. Poincare, Acta Mathematica 13, 1 (1890)] is proposed here. The authors (Recurrences in an isolated quantum many-body system, Science 2018) argue that the theorem should be formulated as “Complex systems return almost exactly into their initial state”. For the first time, this recurrence theorem has been demonstrated with complex quantum multi-particle systems. Our shock-wave model of an earthquake proceeds from the quantum entanglement of protons in hydrogen bonds of lithosphere material. Clearly aftershocks are quantum phenomena which mechanism follows the recurrence theorem.

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2

A. A, Berezin. "The Fermi-Pasta-Ulam Quantum Recurrence in The Dynamics of an Elementary Physical Vacuum Cell and The Problem of its Polarization." Journal of Energy Conservation 1, no. 3 (February 21, 2020): 1–12. http://dx.doi.org/10.14302/issn.2642-3146.jec-20-3179.

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A model of a Quantum recurrence in the dynamics of an elementary physical vacuum cell within the framework of four coupled Shrodinger equations has been suggested. The model of an elementary vacuum cell shows that a Quantum recurrence which represents the dynamics of virtual transformations in the cell, qualitatively differs from that of Poincare and the Fermi-Pasta-Ulam. Whereas these recurrences develop in time or space, the Quantum recurrence develops in a sequence of Fourier images represented by non exponentially separating functions. The sequence experiences random energy additions but no exponential separation occurs. The Quantum recurrence can be defined as the most frequent array of Fourier images that appear in a certain quantum system during a period of its observation. Different scenarios of the Fourier images sequences interpreted as bosons (electron and positron) and fermions (photons) apearing in the solutions of the model demonstrate that during some periods of its observation they become indistinguishable. The quantum dynamics of every physical vacuum cell depends on the dynamics of many other vacuum cells interacting with it, thus the quasi periodicity (during the period of observation) of the Fourier images recurrence can have infinite periods of time and space and the amplitudes of the Fourier images can vary many orders in their magnitudes. Such recurrence times does not correspond even roughly to the Poincare recurrence time of an isolated macroscopic system. It reminds the behavior of spatially coupled standard mappings with different parameters. The amount of energy in the physical vacuum is infinite but extracting a part of it and converting, it into a time-space form requires a process of periodical transfer of the reversible microscopic system dynamics into that of a macroscopic system. This process can be realized through a resonant interaction between the classical and quantum recurrences developing in these two systems. However, a technical realization of this problem is problematic.

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3

Kiss, T., L. Kecskés, M. Štefaňák, and I. Jex. "Recurrence in coined quantum walks." Physica Scripta T135 (July 2009): 014055. http://dx.doi.org/10.1088/0031-8949/2009/t135/014055.

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4

Dhahri, Ameur, and Farrukh Mukhamedov. "Open quantum random walks, quantum Markov chains and recurrence." Reviews in Mathematical Physics 31, no. 07 (July 29, 2019): 1950020. http://dx.doi.org/10.1142/s0129055x1950020x.

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In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution [Formula: see text] of OQRW. This sheds new light on some properties of the measure [Formula: see text]. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of [Formula: see text]-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.

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5

Sikri, A. K., and M. L. Narchal. "Quantum recurrence in a quasibound system." Physical Review A 47, no. 6 (June 1, 1993): 4605–7. http://dx.doi.org/10.1103/physreva.47.4605.

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6

Kryvohuz, Maksym, and Jianshu Cao. "Quantum recurrence from a semiclassical resummation." Chemical Physics 322, no. 1-2 (March 2006): 41–45. http://dx.doi.org/10.1016/j.chemphys.2005.07.021.

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7

Li, Chi Kwong, and Diane Christine Pelejo. "Decomposition of quantum gates." International Journal of Quantum Information 12, no. 01 (February 2014): 1450002. http://dx.doi.org/10.1142/s0219749914500026.

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A recurrence scheme is presented to decompose an n-qubit unitary gate to the product of no more than N(N - 1)/2 single qubit gates with small number of controls, where N = 2n. Detailed description of the recurrence steps and formulas for the number of k-controlled single qubit gates in the decomposition are given. Comparison of the result to a previous scheme is presented, and future research directions are discussed.

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8

Carbone, Raffaella, and Federico Girotti. "Absorption in Invariant Domains for Semigroups of Quantum Channels." Annales Henri Poincaré 22, no. 8 (January 30, 2021): 2497–530. http://dx.doi.org/10.1007/s00023-021-01016-5.

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AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

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9

ISAEV, A. P., and O. OGIEVETSKY. "BRST OPERATOR FOR QUANTUM LIE ALGEBRAS: EXPLICIT FORMULA." International Journal of Modern Physics A 19, supp02 (May 2004): 240–47. http://dx.doi.org/10.1142/s0217751x04020440.

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We continue our study of quantum Lie algebras, an important class of quadratic algebras arising in the Woronowicz calculus on a quantum group. Quantum Lie algebras are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit "quantum" analogues. In particular, there is a BRST operator Q(Q2=0) which generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers a recurrence relation for the operator Q for quantum Lie algebras was given. Here we solve this recurrence relation and obtain an explicit formula for the BRST operator.

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10

Nakanishi, Noboru. "Quantum Recurrence Relation and Its Generating Functions." Publications of the Research Institute for Mathematical Sciences 49, no. 1 (2013): 177–88. http://dx.doi.org/10.4171/prims/101.

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11

Bhattacharyya, Kamal, and Debashis Mukherjee. "On estimates of the quantum recurrence time." Journal of Chemical Physics 84, no. 6 (March 15, 1986): 3212–14. http://dx.doi.org/10.1063/1.450251.

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12

Mukhamedov, Farrukh. "Recurrence and Transience within Quantum Markov Chains." Journal of Physics: Conference Series 819 (March 2017): 012004. http://dx.doi.org/10.1088/1742-6596/819/1/012004.

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13

Nauenberg, M. "Autocorrelation function and quantum recurrence of wavepackets." Journal of Physics B: Atomic, Molecular and Optical Physics 23, no. 15 (August 14, 1990): L385—L390. http://dx.doi.org/10.1088/0953-4075/23/15/001.

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14

POLAVIEJA, GONZALO GARCI A. DE. "Quantum transport, recurrence and localization in H3+." Molecular Physics 87, no. 3 (February 1, 1996): 651–67. http://dx.doi.org/10.1080/00268979650027388.

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15

Chang, Mou-Hsiung. "Recurrence and Transience of Quantum Markov Semigroups." Stochastic Analysis and Applications 33, no. 1 (December 6, 2014): 123–98. http://dx.doi.org/10.1080/07362994.2014.968287.

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16

Fagnola, Franco, and Rolando Rebolledo. "Transience and recurrence of quantum Markov semigroups." Probability Theory and Related Fields 126, no. 2 (June 1, 2003): 289–306. http://dx.doi.org/10.1007/s00440-003-0268-0.

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17

Lozada, A., and P. L. Torres. "Recurrence and coarse-graining in quantum dynamics." Journal of Physics A: Mathematical and General 19, no. 5 (April 1, 1986): L237—L239. http://dx.doi.org/10.1088/0305-4470/19/5/004.

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18

Pandit, Tanmoy, Alaina M. Green, C. Huerta Alderete, Norbert M. Linke, and Raam Uzdin. "Bounds on the recurrence probability in periodically-driven quantum systems." Quantum 6 (April 6, 2022): 682. http://dx.doi.org/10.22331/q-2022-04-06-682.

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Periodically-driven systems are ubiquitous in science and technology. In quantum dynamics, even a small number of periodically-driven spins leads to complicated dynamics. Hence, it is of interest to understand what constraints such dynamics must satisfy. We derive a set of constraints for each number of cycles. For pure initial states, the observable being constrained is the recurrence probability. We use our constraints for detecting undesired coupling to unaccounted environments and drifts in the driving parameters. To illustrate the relevance of these results for modern quantum systems we demonstrate our findings experimentally on a trapped-ion quantum computer, and on various IBM quantum computers. Specifically, we provide two experimental examples where these constraints surpass fundamental bounds associated with known one-cycle constraints. This scheme can potentially be used to detect the effect of the environment in quantum circuits that cannot be classically simulated. Finally, we show that, in practice, testing an n-cycle constraint requires executing only O(n) cycles, which makes the evaluation of constraints associated with hundreds of cycles realistic.

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19

García de Polavieja, Gonzalo, Nicholas G. Fulton, and Jonathan Tennyson. "Quantum transport, recurrence and localization in H+ 3." Molecular Physics 87, no. 3 (February 20, 1996): 651–67. http://dx.doi.org/10.1080/00268979600100451.

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20

Sikri, A. K., and M. L. Narchal. "Partial quantum recurrence in free and quasibound systems." Pramana 52, no. 5 (May 1999): 453–57. http://dx.doi.org/10.1007/bf02830092.

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21

Štefaňák, M., T. Kiss, and I. Jex. "Recurrence of biased quantum walks on a line." New Journal of Physics 11, no. 4 (April 23, 2009): 043027. http://dx.doi.org/10.1088/1367-2630/11/4/043027.

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22

WU, SHAO-XIONG, JUN ZHANG, CHANG-SHUI YU, and HE-SHAN SONG. "QUANTUM CORRELATIONS IN THE ENTANGLEMENT DISTILLATION PROTOCOLS." International Journal of Quantum Information 11, no. 03 (April 2013): 1350029. http://dx.doi.org/10.1142/s0219749913500299.

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We study the quantum correlations between source and target pairs in different protocols of entanglement distillation of one kind of entangled states. We find that there does not exist any quantum correlation in the standard recurrence distillation protocol, while quantum discord and even quantum entanglement are always present in the other two cases of the improved distillation protocols. In the three cases, the distillation efficiency improved with the quantum correlations enhanced.

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23

李, 宗谚. "Quantum Properties of Yin-Yang Recurrence of Chinese Medicine." Traditional Chinese Medicine 06, no. 02 (2017): 40–45. http://dx.doi.org/10.12677/tcm.2017.62008.

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24

Main, Jörg, Vladimir A. Mandelshtam, and Howard S. Taylor. "High Resolution Quantum Recurrence Spectra: Beyond the Uncertainty Principle." Physical Review Letters 78, no. 23 (June 9, 1997): 4351–54. http://dx.doi.org/10.1103/physrevlett.78.4351.

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25

Nitsche, Thomas, Sonja Barkhofen, Regina Kruse, Linda Sansoni, Martin Štefaňák, Aurél Gábris, Václav Potoček, Tamás Kiss, Igor Jex, and Christine Silberhorn. "Probing measurement-induced effects in quantum walks via recurrence." Science Advances 4, no. 6 (June 2018): eaar6444. http://dx.doi.org/10.1126/sciadv.aar6444.

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26

Laha, Pradip, S. Lakshmibala, and V. Balakrishnan. "Recurrence network analysis in a model tripartite quantum system." EPL (Europhysics Letters) 125, no. 6 (May 2, 2019): 60005. http://dx.doi.org/10.1209/0295-5075/125/60005.

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27

Grünbaum, F. A., C. F. Lardizabal, and L. Velázquez. "Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules." Annales Henri Poincaré 21, no. 1 (November 13, 2019): 189–239. http://dx.doi.org/10.1007/s00023-019-00863-7.

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28

Jacq, Thomas S., and Carlos F. Lardizabal. "Homogeneous open quantum walks on the line: criteria for site recurrence and absorption." Quantum Information and Computation 21, no. 1&2 (February 2021): 0037–58. http://dx.doi.org/10.26421/qic21.1-2-3.

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In this work, we study open quantum random walks, as described by S. Attal et al.. These objects are given in terms of completely positive maps acting on trace-class operators, leading to one of the simplest open quantum versions of the recurrence problem for classical, discrete-time random walks. This work focuses on obtaining criteria for site recurrence of nearest-neighbor, homogeneous walks on the integer line, with the description presented here making use of recent results of the theory of open walks, most particularly regarding reducibility properties of the operators involved. This allows us to obtain a complete criterion for site recurrence in the case for which the internal degree of freedom of each site (coin space) is of dimension 2. We also present the analogous result for irreducible walks with an internal degree of arbitrary finite dimension and the absorption problem for walks on the semi-infinite line.

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29

Berezin, A. A. "Das an Electric Current have an Acoustic Component?" Journal of Energy Conservation 1, no. 2 (March 9, 2019): 1–14. http://dx.doi.org/10.14302/issn.2642-3146.jec-19-2663.

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The quantum model of electric current suggested by Feynman has been enlarged by n difference-differential Hamiltonian equations describing the phonon dynamics in one dimensional crystallyne lattice. The process of interaction between the electron and phonon components in a crystalline lattice of a conductor has been described by 2n parametrically coupled difference-differential Hamiltonian equations. Computer analysis of the system of these coupled equations showed that their solutions represent a form of the quantum recurrence similar to the Fermi-Pasta-Ulam recurrence. The results of the research might reconsider the existing concept of electric current and will be possibly helpful in developing an acoustic «laser».

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30

AURICH, R., and F. STEINER. "TEMPORAL QUANTUM CHAOS." International Journal of Modern Physics B 13, no. 18 (July 20, 1999): 2361–69. http://dx.doi.org/10.1142/s0217979299002459.

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We study the long-time behavior of bound quantum systems whose classical dynamics is chaotic and put forward two conjectures. Conjecture A states that the autocorrelation function C(t)=<Ψ(0)|Ψ(t)> of a delocalized initial state |Ψ(0)> shows characteristic fluctuations, which we identify with a universal signature of temporal quantum chaos. For example, for the (appropriately normalized) value distribution of S~|C(t)| we predict the distribution P(S)=(π/2)Se-πS2/4. Conjecture B gives the best possible upper bound for a generalized Weyl sum and is related to the extremely large recurrence times in temporal quantum chaos. Numerical tests carried out for numerous chaotic systems confirm nicely the two conjectures and thus provide strong evidence for temporal quantum chaos.

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31

Kocharovsky, Vitaly, Vladimir Kocharovsky, Vladimir Martyanov, and Sergey Tarasov. "Exact Recursive Calculation of Circulant Permanents: A Band of Different Diagonals inside a Uniform Matrix." Entropy 23, no. 11 (October 28, 2021): 1423. http://dx.doi.org/10.3390/e23111423.

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We present a finite-order system of recurrence relations for the permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k=1,2 and 3) and the method for deriving such recurrence relations, which is based on the permanents of the matrices with defects. The proposed system of linear recurrence equations with variable coefficients provides a powerful tool for the analysis of the circulant permanents, their fast, linear-time computing; and finding their asymptotics in a large-matrix-size limit. The latter problem is an open fundamental problem. Its solution would be tremendously important for a unified analysis of a wide range of the nature’s ♯P-hard problems, including problems in the physics of many-body systems, critical phenomena, quantum computing, quantum field theory, theory of chaos, fractals, theory of graphs, number theory, combinatorics, cryptography, etc.

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32

Konno, Norio, and Etsuo Segawa. "Localization of quantum walks via the CGMV method." Quantum Information and Computation 11, no. 5&6 (May 2011): 485–96. http://dx.doi.org/10.26421/qic11.5-6-9.

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We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle, expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al.

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33

HEGSTROM, ROGER A., and GWEN ADSHEAD. "Incompatible Variables and “Quantal” Phenomena in Psychology." Journal of North Carolina Academy of Science 127, no. 1 (March 1, 2011): 18–27. http://dx.doi.org/10.7572/2167-5880-127.1.18.

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Abstract Mindfulness-based cognitive therapy (MBCT) is an area of modern psychology that provides a successful method for preventing the recurrence of depression. The standard theory of MBCT may be interpreted in terms of a simple theoretical model that employs incompatible variables as its fundamental observable quantities. Although the MBCT theory is not related to or derived from quantum theory, the existence of incompatible variables results in “quantal” phenomena, such as interference and the uncertainty principle, which have been widely believed to occur only as a consequence of the laws of quantum mechanics. The predictions of the model are subject to experimental testing and at present the model agrees with all published clinical data. To our knowledge, this work is the first quantitative theoretical treatment of a natural non-quantal system possessing incompatible variables. It confirms the intuition of Bohr and Heisenberg that incompatible variables may exist in human psychology.

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34

Barhoumi, Abdessatar, and Abdessatar Souissi. "Recurrence of a class of quantum Markov chains on trees." Chaos, Solitons & Fractals 164 (November 2022): 112644. http://dx.doi.org/10.1016/j.chaos.2022.112644.

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35

Sudheesh, C., S. Lakshmibala, and V. Balakrishnan. "Recurrence statistics of observables in quantum-mechanical wave packet dynamics." EPL (Europhysics Letters) 90, no. 5 (June 1, 2010): 50001. http://dx.doi.org/10.1209/0295-5075/90/50001.

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36

Nitta, Hiroya, Mitsuo Shoji, Masahiro Takahata, Masayoshi Nakano, Daisuke Yamaki, and Kizashi Yamaguchi. "Quantum dynamics of exciton recurrence motion in dendritic molecular aggregates." Journal of Photochemistry and Photobiology A: Chemistry 178, no. 2-3 (March 2006): 264–70. http://dx.doi.org/10.1016/j.jphotochem.2005.10.040.

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37

Bourgain, J., F. A. Grünbaum, L. Velázquez, and J. Wilkening. "Quantum Recurrence of a Subspace and Operator-Valued Schur Functions." Communications in Mathematical Physics 329, no. 3 (March 19, 2014): 1031–67. http://dx.doi.org/10.1007/s00220-014-1929-9.

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38

Dere, Paçin. "Some recurrence formulas for the q-Bernoulli and q-Euler polynomials." Filomat 34, no. 2 (2020): 663–69. http://dx.doi.org/10.2298/fil2002663d.

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The recurrence relations have a very important place for the special polynomials such as q-Appell polynomials. In this paper, we give some recurrence formulas that allow us a better understanding of q-Appell polynomials. We investigate the q-Bernoulli polynomials and q-Euler polynomials, which are q-Appell polynomials, and we obtain their recurrence formulas by using the methods of the q-umbral calculus and the quantum calculus. Our methods include some operators which are quite handy for obtaining relations for the q-Appell polynomials. Especially, some applications of q-derivative operator are used in this work.

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39

Genoni, Alessandro. "On the use of the Obara–Saika recurrence relations for the calculation of structure factors in quantum crystallography." Acta Crystallographica Section A Foundations and Advances 76, no. 2 (February 11, 2020): 172–79. http://dx.doi.org/10.1107/s205327332000042x.

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Modern methods of quantum crystallography are techniques firmly rooted in quantum chemistry and, as in many quantum chemical strategies, electron densities are expressed as two-centre expansions that involve basis functions centred on atomic nuclei. Therefore, the computation of the necessary structure factors requires the evaluation of Fourier transform integrals of basis function products. Since these functions are usually Cartesian Gaussians, in this communication it is shown that the Fourier integrals can be efficiently calculated by exploiting an extension of the Obara–Saika recurrence formulas, which are successfully used by quantum chemists in the computation of molecular integrals. Implementation and future perspectives of the technique are also discussed.

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40

Syros, C. "Boltzmann and Zermelo Versus Loschmidt and Poincaré — is there any Recurrence?" International Journal of Modern Physics B 12, no. 27n28 (November 10, 1998): 2785–801. http://dx.doi.org/10.1142/s0217979298001629.

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It is shown that Poincaré's recurrence theorem incorporates Loschmidt's requirement for velocity reversion in thermodynamic gas systems. It differs essentially from Hamiltonian dynamics from which Boltzmann's H-theorem follows. The inverse automorphism, T-1, on which is based the demonstration of the recurrence theorem does not exist for atoms and molecule systems. Thermodynamic systems need not spontaneously return to states they occupied in the past and a Zermelo paradox has never existed for them. The same conclusion follows a fortiori for quantum systems in chrono-topology. Poincaré's recurrence theorem does not conflict with Boltzmann's H-theorem because they apply to systems described by quite different mathematical structures.

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41

Kaminishi, Eriko, Jun Sato, and Tetsuo Deguchi. "Recurrence Time in the Quantum Dynamics of the 1D Bose Gas." Journal of the Physical Society of Japan 84, no. 6 (June 15, 2015): 064002. http://dx.doi.org/10.7566/jpsj.84.064002.

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42

Segawa, Etsuo. "Localization of Quantum Walks Induced by Recurrence Properties of Random Walks." Journal of Computational and Theoretical Nanoscience 10, no. 7 (July 1, 2013): 1583–90. http://dx.doi.org/10.1166/jctn.2013.3092.

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43

Lardizabal, Carlos F., and Rafael R. Souza. "On a Class of Quantum Channels, Open Random Walks and Recurrence." Journal of Statistical Physics 159, no. 4 (February 17, 2015): 772–96. http://dx.doi.org/10.1007/s10955-015-1217-x.

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44

Gimeno, V., and J. M. Sotoca. "Upper bounds for the Poincaré recurrence time in quantum mixed states." Journal of Physics A: Mathematical and Theoretical 50, no. 18 (April 5, 2017): 185302. http://dx.doi.org/10.1088/1751-8121/aa67fe.

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45

Mizrahi, S. S., M. H. Y. Moussa, and D. Otero. "Recurrence and decoherence times of quantum states in a measurement process." Physics Letters A 180, no. 3 (September 1993): 244–48. http://dx.doi.org/10.1016/0375-9601(93)90704-4.

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46

BURDÍK, Č., and O. NAVRÁTIL. "THE q-BOSON REALIZATIONS OF THE QUANTUM GROUP Uq(Cn)." International Journal of Modern Physics A 14, no. 28 (November 10, 1999): 4491–500. http://dx.doi.org/10.1142/s0217751x99002104.

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We give explicit realization for the quantum enveloping algebras Uq(Cn). To obtain recurrence formulae we extend simply the algebra Uq(Cn) to [Formula: see text], for which Uq(Cn) is a subalgebra. In these formulae the generators of the algebra are expressed by means of 2n-1 canonical q-boson pairs and one auxiliary representation of [Formula: see text].

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47

Liashyk, A., and S. Z. Pakuliak. "Recurrence relations for off-shell Bethe vectors in trigonometric integrable models." Journal of Physics A: Mathematical and Theoretical 55, no. 7 (January 25, 2022): 075201. http://dx.doi.org/10.1088/1751-8121/ac491b.

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Abstract The zero modes method is applied in order to get the action of the monodromy matrix entries on off-shell Bethe vectors in quantum integrable models associated with U q ( g l N ) -invariant R-matrices. The action formulas allow to get recurrence relations for off-shell Bethe vectors and for highest coefficients of the Bethe vectors scalar product.

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48

Khavkine, Igor. "Recurrence relation for the 6j-symbol of suq(2) as a symmetric eigenvalue problem." International Journal of Geometric Methods in Modern Physics 12, no. 10 (October 25, 2015): 1550117. http://dx.doi.org/10.1142/s0219887815501170.

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A well-known recurrence relation for the 6j-symbol of the quantum group su q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in Appendix A. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q = 1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley–Lieb recoupling theory to simplify intermediate calculations.

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49

de Jong, Jins, Alexander Hock, and Raimar Wulkenhaar. "Nested Catalan tables and a recurrence relation in noncommutative quantum field theory." Annales de l’Institut Henri Poincaré D 9, no. 1 (April 11, 2022): 47–72. http://dx.doi.org/10.4171/aihpd/113.

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50

Solodukhin, Sergey N. "Renormalization group equations and the recurrence pole relations in pure quantum gravity." Nuclear Physics B 962 (January 2021): 115246. http://dx.doi.org/10.1016/j.nuclphysb.2020.115246.

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