Geodesic
Quantization scheme for modular $q$-difference equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 3, pp. 500-509 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider modular pairs of certain second-order $q$-difference equations. An example of such a pair is the $t$-$Q$ Baxter equations for the quantum relativistic Toda lattice in the strong coupling regime. Another example from quantum mechanics is $q$-deformation of the Schrödinger equation with a hyperbolic potential. We show that the analyticity condition for the wave function or the Baxter function leads to a set of transcendental equations for the coefficients of the potential or the transfer matrix, the solution of which is their discrete spectrum.
Keywords: Baxter equations, modular dualization, strong coupling regime. 
@article{TMF_2005_142_3_a2,
     author = {S. M. Sergeev},
     title = {Quantization scheme for modular $q$-difference equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {500--509},
     year = {2005},
     volume = {142},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a2/}
}
TY  - JOUR
AU  - S. M. Sergeev
TI  - Quantization scheme for modular $q$-difference equations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2005
SP  - 500
EP  - 509
VL  - 142
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a2/
LA  - ru
ID  - TMF_2005_142_3_a2
ER  - 
%0 Journal Article
%A S. M. Sergeev
%T Quantization scheme for modular $q$-difference equations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2005
%P 500-509
%V 142
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a2/
%G ru
%F TMF_2005_142_3_a2
S. M. Sergeev. Quantization scheme for modular $q$-difference equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 3, pp. 500-509. http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a2/

[1] L. D. Faddeev, Lett. Math. Phys., 34 (1995), 249–254 ; L. D. Faddeev, “Modular double of quantum group”, Quantization, Deformation and Symmetries, Conférence Moshé Flato 1999. V. 1 (Dijon, France, September 5–8, 1999), Math. Phys. Stud., 21, eds. G. Dito, D. Sternheimer, Kluwer, Dordrecht, 2000, 149–156 ; L. D. Faddeev, R. M. Kashaev, A. Yu. Volkov, Commun. Math. Phys., 219 (2001), 199–219 ; L. D. Faddeev, R. M. Kashaev, J. Phys. A, 35 (2002), 4043–4048 | DOI | MR | Zbl | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[2] S. Kharchev, D. Lebedev, M. Semenov-Tian-Shansky, Commun. Math. Phys., 225 (2002), 573–609 | DOI | MR | Zbl

[3] M. Gutzwiller, Ann. of Phys., 133 (1981), 304–331 ; V. Pasquier, M. Gaudin, J. Phys. A, 25 (1992), 5243–5252 ; S. Kharchev, D. Lebedev, Lett. Math. Phys., 50 (1999), 53–77 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[4] V. Kuznetsov, A. Tsiganov, Separation of variables for the quantum relativistic Toda lattices, ; G. Pronko, S. Sergeev, J. Appl. Math., 1 (2001), 47–68 E-print hep-th/9402111 | MR | DOI | MR | Zbl

[5] F. A. Smirnov, Dual Baxter equations and quantization of Affine Jacobian, E-print math-ph/0001032 | MR

[6] L. D. Faddeev, R. M. Kashaev, Mod. Phys. Lett. A, 9 (1994), 427–434 | DOI | MR | Zbl

[7] S. M. Sergeev, J. Phys. A, 32 (1999), 5693–5714 ; С. М. Сергеев, ТМФ, 124:3 (2000), 391–409 | DOI | MR | Zbl | DOI | MR | Zbl