Geodesic
Reduction of quasipotential equations to Sturm–Liouville problems and the comparison equation method
Teoretičeskaâ i matematičeskaâ fizika, Tome 48 (1981) no. 1, pp. 80-88 Cet article a éte moissonné depuis la source Math-Net.Ru

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The quasipotential equation that describes the interaction of two scalar particles of equal mass through a quasipotential of the form $V(r)=-gr^{-1}$ in the coordinate space is reduced to the corresponding Sturm–Liouville problem in the momentum space. An iterative comparison equation method is formulated in general form and makes it possible to find the solutions of such problems as asymptotic series in reciprocal powers of the coupling constant. The method is used to obtain the discrete spectrum at all binding energies. The problem of the equality of the solutions of the comparison equation method and the exact solutions at singular points of the original equation is investigated for the example of a modified variant of the Sturm–Liouville quasipotential problem. Such an investigation is needed if the original equation does not admit exact comparison. 
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     title = {Reduction of quasipotential equations to {Sturm{\textendash}Liouville} problems and the comparison equation method},
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V. Sh. Gogokhiya. Reduction of quasipotential equations to Sturm–Liouville problems and the comparison equation method. Teoretičeskaâ i matematičeskaâ fizika, Tome 48 (1981) no. 1, pp. 80-88. http://geodesic.mathdoc.fr/item/TMF_1981_48_1_a8/

[1] Logunov A. A., Tavkhelidze A. N., “Quasi-optical approach in quantum fild theory”, Nuovo Cim., 29:2 (1963), 380–399 | DOI | MR

[2] Arbuzov B. A., Filippov A. T., “Singular potentials in the Lippman-Schwinger equation”, Phys. Lett., 13:1 (1964), 95–96 | DOI | MR

[3] Gogokhiya V. Sh., Filippov A. T., “Singulyarnoe kvazipotentsialnoe uravnenie”, TMF, 21:1 (1974), 37–48 | MR | Zbl

[4] Gogokhiya V. Sh., Mavlo D. P., Filippov A. T., O reshenii relyativistskikh kvazipotentsialnykh uravnenii dlya svyazannykh sostoyanii, Preprint R2-9893, OIYaI, Dubna, 1976

[5] Gogokhiya V. Sh., Mavlo D. P., Filippov A. T., “Reshenie singulyarnogo kvazipotentsialnogo uravneniya dlya svyazannykh sostoyanii”, TMF, 27:3 (1976), 323–336

[6] Cherry T. M., “Uniform asymptotic formulae for functions with transition points”, Trans. Amer. Math. Soc., 68:1 (1950), 224–257 | DOI | MR | Zbl

[7] Langer R. A., “On the asymptotic solutions of ordinary differential equations with an application to the Bessel functions of large order”, Trans. Amer. Math. Soc., 33:1 (1931), 23–64 | DOI | MR | Zbl

[8] Olver F. W., “The asymptotic solution of linear differential equations of the second order for large values of a parameter”, Phil. Trans. Roy. Soc. London, A247:922 (1954), 307–327 | DOI | MR

[9] Ponomarev L. I., Lektsii po kvaziklassike, Preprint 67-53, ITF, Kiev, 1968 | MR

[10] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, Nauka, M., 1965 | MR

[11] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971 | MR

[12] Gogokhiya V. Sh., “Relyativistskaya kulonovskaya zadacha v kvazipotentsialnoi formulirovke”, Soobscheniya AN GrSSR, 90:2 (1978), 333–336

[13] Case K. M., “Singular potentials”, Phys. Rev., 80:5 (1950), 797–806 | DOI | MR | Zbl

[14] Perelomov A. M., Popov V. S., “«Padenie na tsentr» v kvantovoi mekhanike”, TMF, 4:1 (1970), 48–65 | MR

[15] Langer R. E., “On the connection formulas and the solution of the wave equation”, Phys. Rev., 51:8 (1937), 669–676 | DOI | Zbl