Geodesic
On power series representing solutions of the one-dimensional time-independent Schrödinger equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 6, pp. 973-984 
N. P. Trotsenko. On power series representing solutions of the one-dimensional time-independent Schrödinger equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 6, pp. 973-984. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_6_a5/
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     author = {N. P. Trotsenko},
     title = {On power series representing solutions of the one-dimensional time-independent {Schr\"odinger} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {973--984},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_6_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For the equation $\chi''(x)=u(x)\chi(x)$ with infinitely smooth $u(x)$, the general solution $\chi(x)$ is found in the form of a power series. The coefficients of the series are expressed via all derivatives $u^{(m)}(y)$ of the function $u(x)$ at a fixed point $y$. Examples of solutions for particular functions $u(x)$ are considered.

[1] Vinogradov A. V., Popov A. V., Prokopovich D. V., “O yavnom parametricheskom opisanii voln v periodicheskikh sredakh”, Zh. vychisl. matem. i matem. fiz., 49:6 (2009), 1119–1130 | Zbl

[2] Kovalev M. D., “Chislo energeticheskikh urovnei chastitsy v grebenchatoi strukture”, Zh. vychisl. matem. i matem. fiz., 47:9 (2007), 1557–1575

[3] Maier A. A., Kovalev M. D., “Dispersionnoe uravnenie dlya sobstvennykh znachenii effektivnogo pokazatelya prelomleniya v mnogosloinoi volnovodnoi strukture”, Dokl. AN, 407:6 (2006), 766–769 | Zbl

[4] Kovalev M. D., “Ob energeticheskikh urovnyakh chastitsy v grebenchatoi strukture”, Dokl. AN, 419:6 (2008), 749–753 | Zbl

[5] Sobko A. A., “Vychislenie sobstvennykh znachenii diskretnogo spektra uravneniya Shredingera dlya proizvolnykh odnomernykh, finitnykh, chetnykh potentsialnykh yam”, Dokl. AN, 414:4 (2007), 464–468 | Zbl

[6] Komarov D. A., Morev S. P., Darmaev A. N., Ryadnov A. Yu., “Chislennoe reshenie uravneniya Shredingera dlya elektrona v potentsialnoi yame proizvolnoi formy”, Radiotekhnika i elektronika, 59:8 (2014), 799–803 | DOI

[7] Masoero D., “Poles of integrale tritronquee and anharmonic oscillators. A WKB approach”, J. Phys. A: Math. Theor., 43:9 (2010), 095201 | DOI | MR | Zbl

[8] Dorey P., Tateo R., “On the relation between Stokes multipliers and the TQ systems of conformal field theory”, Nucl. Phys., B563 (1999), 573–602 | DOI | MR | Zbl

[9] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1976