Geodesic
Algebraic calculation of the resolvent of a generalized quantum oscillator in a space of dimension $D$
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 1, pp. 109-117 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the formalism based on using the $sl(2)$ algebra instead of the conventional Heisenberg algebra for isotropic models of quantum mechanics. The operators of the squared momentum $p^2$ and squared coordinates $q^2$ and also the dilation operator $H=i(pq+qp)$ are used as its generators. This allows calculating with the space dimension $D$ as an arbitrary, not necessarily integer parameter. We obtain integral representations for the resolvent and its trace for a generalized harmonic oscillator with the Hamiltonian $H(a,b,c)=ap^2+bq^2+cH$ and any $D$ and study their analytic properties for different model parameter values.
Keywords: generalized quantum oscillator, $sl(2)$ algebra, isotropic model of quantum mechanics, resolvent
Mots-clés : spectral decomposition. 
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K. S. Karpov; Yu. M. Pis'mak. Algebraic calculation of the resolvent of a generalized quantum oscillator in a space of dimension $D$. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 1, pp. 109-117. http://geodesic.mathdoc.fr/item/TMF_2015_185_1_a9/

[1] D. Kreimer, Phys. Rep., 363:4–6 (2002), 387–424, arXiv: ; D. Kreimer, Knots and Feynman Diagrams, Cambridge Lecture Notes in Physics, 13, Cambridge Univ. Press, Cambridge, 2000 hep-th/0010059 | DOI | MR | Zbl | MR

[2] S. Moch, P. Uwer, S. Weinzierl, J. Math. Phys., 43:6 (2002), 3363–3386, arXiv: hep-ph/0110083 | DOI | MR | Zbl

[3] A. I. Davydychev, R. Delbourgo, J. Math. Phys., 39:9 (1998), 4299–4334, arXiv: hep-th/9709216 | DOI | MR | Zbl

[4] D. J. Broadhurst, Eur. Phys. J. C, 8:2 (1999), 311–333 | DOI | MR

[5] A. I. Davydychev, M. Yu. Kalmykov, Nucl. Phys. B, 605:1–3 (2001), 266–318 | DOI | MR | Zbl

[6] V. A. Smirnov, Nucl. Phys. B: Proc. Suppl., 116 (2003), 417–421, arXiv: ; Phys. Lett. B, 547:3–4 (2002), 239–244, arXiv: ; 500:3–4 (2001), 330–337, arXiv: ; 460:3–4 (1999), 397–404 hep-ph/0209295hep-ph/0209193hep-ph/0011056 | DOI | Zbl | DOI | Zbl | DOI | MR | Zbl | DOI

[7] A. N. Vasilev, Kvantovopolevaya renormgruppa v teorii kriticheskogo povedeniya i stokhasticheskoi dinamike, Izd-vo PIYaF, SPb., 1998 | MR | Zbl

[8] F. V. Tkachov, Phys. Lett. B, 100:1 (1981), 65–68 | DOI | MR

[9] A. N. Vasilev, Yu. M. Pismak, Yu. R. Khonkonen, TMF, 47:3 (1981), 291–306 | DOI

[10] A. Zamolodchikov, Phys. Lett. B, 97:1 (1980), 63–66 | DOI | MR

[11] A. P. Isaev, Nucl. Phys. B, 662:3 (2003), 461–475, arXiv: hep-th/0303056 | DOI | MR | Zbl

[12] E. Ch. Titchmarsh, Razlozhenie po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, 1 chast, IL, M., 1960 | MR | Zbl

[13] V. L. Bakhrakh, S. I. Vetchinkin, TMF, 6:3 (1971), 392–402 | DOI | MR

[14] V. L. Bakhrakh, S. I. Vetchinkin, S. V. Khristenko, TMF, 12:2 (1972), 223–226 | DOI | Zbl

[15] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1963 | MR | Zbl