Geodesic
Schrödinger potentials solvable in terms of the confluent Heun
Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 1, pp. 20-35 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We show that if the potential is proportional to an energy-independent continuous parameter, then there exist 15 choices for the coordinate transformation that provide energy-independent potentials whose shape is independent of that parameter and for which the one-dimensional stationary Schrödinger equation is solvable in terms of the confluent Heun functions. All these potentials are also energy-independent and are determined by seven parameters. Because the confluent Heun equation is symmetric under transposition of its regular singularities, only nine of these potentials are independent. Five of the independent potentials are different generalizations of either a hypergeometric or a confluent hypergeometric classical potential, one potential as special cases includes potentials of two hypergeometric types (the Morse confluent hypergeometric and the Eckart hypergeometric potentials), and the remaining three potentials include five-parameter conditionally integrable confluent hypergeometric potentials. Not one of the confluent Heun potentials, generally speaking, can be transformed into any other by a parameter choice.
Keywords: stationary Schrödinger equation, integrable potential
Mots-clés : confluent Heun equation. 
@article{TMF_2016_188_1_a1,
     author = {A. M. Ishkhanyan},
     title = {Schr\"odinger potentials solvable in terms of the~confluent {Heun}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {20--35},
     year = {2016},
     volume = {188},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_188_1_a1/}
}
TY  - JOUR
AU  - A. M. Ishkhanyan
TI  - Schrödinger potentials solvable in terms of the confluent Heun
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 20
EP  - 35
VL  - 188
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_188_1_a1/
LA  - ru
ID  - TMF_2016_188_1_a1
ER  - 
%0 Journal Article
%A A. M. Ishkhanyan
%T Schrödinger potentials solvable in terms of the confluent Heun
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 20-35
%V 188
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2016_188_1_a1/
%G ru
%F TMF_2016_188_1_a1
A. M. Ishkhanyan. Schrödinger potentials solvable in terms of the confluent Heun. Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 1, pp. 20-35. http://geodesic.mathdoc.fr/item/TMF_2016_188_1_a1/

[1] A. Ronveaux (ed.), Heun's Differential Equations, Oxford Univ. Press, Oxford, 1995 | MR | Zbl

[2] S. Yu. Slavyanov, V. Lai, Spetsialnye funktsii: edinaya teoriya, osnovannaya na analize singulyarnostei, Nevskii dialekt, SPb., 2002 | MR

[3] I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Sferoidalnye i kulonovskie sferoidalnye funktsii, Nauka, M., 1976 | MR

[4] E. Schrödinger, Ann. Phys., 384:4 (1926), 361–376 | DOI

[5] P. M. Morse, Phys. Rev., 34:1 (1929), 57–64 | DOI | Zbl

[6] C. Eckart, Phys. Rev., 35:11 (1930), 1303–1309 | DOI | Zbl

[7] N. Rosen, P. M. Morse, Phys. Rev., 42:2 (1932), 210–217 | DOI

[8] R. D. Woods, D. S. Saxon, Phys. Rev., 95:2 (1954), 577–578 | DOI

[9] M. F. Manning, N. Rosen, Phys. Rev., 44:11 (1933), 953 | DOI

[10] L. Hulthén, Ark. Mat. Astr. Fys. A, 28:5 (1942), 1–12 | MR | Zbl

[11] G. Pöschl, E. Teller, Z. Phys. A: Hadrons and Nuclei, 83:3–4 (1933), 143–151 | DOI | Zbl

[12] F. Scarf, Phys. Rev., 112:4 (1958), 1137–1140 | DOI | MR | Zbl

[13] A. Kratzer, Z. Phys., 3:5 (1920), 289–307 | DOI

[14] J. H. Lambert, Acta Helvetica, 3 (1758), 128–168

[15] L. Euler, Acta Acad. Scient. Petropol., 2 (1783), 29–51

[16] A. K. Bose, Phys. Lett., 7:4 (1963), 245–246 | DOI | MR

[17] G. A. Natanzon, TMF, 38:2 (1979), 219–229 | DOI | MR | Zbl

[18] A. Lemieux, A. K. Bose, Ann. Inst. H. Poincaré Sect. A (N. S.), 10:4 (1969), 259–270 | MR

[19] E. W. Leaver, J. Math. Phys., 27:5 (1986), 1238–1265 | DOI | MR | Zbl

[20] B. W. Williams, Phys. Lett. A, 334:2–3 (2005), 117–122 | DOI | MR | Zbl

[21] A. M. Ishkhanyan, Phys. Lett. A, 380:5–6 (2016), 640–644, arXiv: 1509.00846 | DOI | MR | Zbl

[22] A. M. Ishkhanyan, Eur. Phys. Lett., 112:1 (2015), 10006 | DOI

[23] M. F. Manning, Phys. Rev., 48:2 (1935), 161–164 | DOI | Zbl

[24] L. J. El-Jaick, B. D. B. Figueiredo, J. Math. Phys., 49:8 (2008), 083508, 28 pp. | DOI | MR | Zbl

[25] D. Batic, R. Williams, M. Nowakowski, J. Phys. A: Math. Theor., 46:24 (2013), 245204, 21 pp. | DOI | MR | Zbl

[26] A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, TMF, 88:1 (1991), 59–65 | MR | Zbl

[27] A. M. Ishkhanyan, A. E. Grigoryan, J. Phys. A: Math. Theor., 47:46 (2014), 465205, 22 pp. | DOI | MR | Zbl

[28] C. Quesne, J. Phys. A: Math. Theor., 41:39 (2008), 392001, 6 pp. | DOI | MR | Zbl

[29] S. Odake, R. Sasaki, Phys. Lett. B, 679:4 (2009), 414–417, arXiv: 0906.0142 | DOI | MR

[30] A. M. Ishkhanyan, T. A. Shahverdyan, T. A. Ishkhanyan, Eur. Phys. J. D, 69:1 (2015), 10, 14 pp., arXiv: 1404.3922 | DOI

[31] T. A. Shahverdyan, T. A. Ishkhanyan, A. E. Grigoryan, A. M. Ishkhanyan, J. Contemp. Phys. (Armenian Acad. Sci.), 50:3 (2015), 211–226, arXiv: 1412.1378 | DOI

[32] G. Junker, P. Roy, Phys. Lett. A, 232:3–4 (1997), 155–161 | MR | Zbl

[33] G. Lévai, P. Roy, Phys. Lett. A, 264:2–3 (1999), 117–123, arXiv: quant-ph/9909004 | MR

[34] A. Sinha, G. Lévai, P. Roy, Phys. Lett. A, 322:1–2 (2004), 78–83 | DOI | MR | Zbl

[35] A. Ya. Kazakov, S. Yu. Slavyanov, TMF, 179:2 (2014), 189–195 | DOI | MR | Zbl