Geodesic
Inverse scattering problem for the Schrödinger equation with an additional quadratic potential on the entire axis
Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 1, pp. 54-63 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Schrödinger equation with an additional quadratic potential on the entire axis and use the transformation operator method to study the direct and inverse problems of the scattering theory. We obtain the main integral equations of the inverse problem and prove that the basic equations are uniquely solvable.
Keywords: Schrödinger equation, oscillator, inverse scattering problem, basic equation.
Mots-clés : reflection coefficient 
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     title = {Inverse scattering problem for {the~Schr\"odinger} equation with an~additional quadratic potential on the~entire axis},
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I. M. Guseinov; A. Kh. Khanmamedov; A. F. Mamedova. Inverse scattering problem for the Schrödinger equation with an additional quadratic potential on the entire axis. Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 1, pp. 54-63. http://geodesic.mathdoc.fr/item/TMF_2018_195_1_a4/

[1] F. A. Berezin, M. A. Shubin, Uravnenie Shredingera, Izd-vo Mosk. un-ta, M., 1983 | MR

[2] H. P. McKean, E. Trubowitz, “The spectral class of the quantum-mechanical harmonic oscillator”, Commun. Math. Phys., 82:4 (1982), 471–495 | DOI | MR | Zbl

[3] B. M. Levtan, “Ob operatorakh Shturma–Liuvillya na vsei pryamoi s odinakovym diskretnym spektrom”, Matem. sb., 132(174):1 (1987), 73–103 | DOI | MR | Zbl

[4] D. Chelkak, P. Kargaev, E. Korotyaev, “Inverse problem for harmonic oscillator perturbed by potential, characterization”, Commun. Math. Phys., 249:1 (2004), 133–196 | DOI | MR | Zbl

[5] L. D. Faddeev, “Obratnaya zadacha kvantovoi teorii rasseyaniya”, UMN, 14:4 (1959), 57–119 | DOI | MR | Zbl

[6] L. D. Faddeev, “Svoistva $S$-matritsy odnomernogo uravneniya Shredingera”, Tr. MIAN SSSR, 73 (1964), 314–326 | MR | Zbl

[7] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR

[8] M. G. Gasymov, B. A. Mustafaev, “Obratnaya zadacha rasseyaniya dlya angarmonicheskogo uravneniya na poluosi”, Dokl. AN SSSR, 228:1 (1976), 11–14 | MR | Zbl

[9] Y. Li, “One special inverse problem of the second order differential equation on the whole real axis”, Chinese Ann. Math., 2:2 (1981), 147–155 | MR

[10] A. P. Kachalov, Ya. V. Kurylev, “Metod operatorov preobrazovaniya v obratnoi zadache rasseyaniya. Odnomernyi shtark-effekt”, Zap. nauchn. sem. LOMI, 179 (1989), 73–87 | DOI | MR | Zbl

[11] P. P. Kulish, “Obratnaya zadacha rasseyaniya dlya uravneniya Shredingera na osi”, Matem. zametki, 4:6 (1968), 677–684 | DOI | MR | Zbl

[12] Y. Liu, “Scattering and spectral theory for Stark Hamiltonians in one dimension”, Math. Scand., 72:2 (1993), 265–297 | DOI | MR | Zbl

[13] E. A. Kuznetsov, A. V. Mikhailov, “Ustoichivost statsionarnykh voln v nelineinykh sredakh so slaboi dispersiei”, ZhETF, 67:5 (1974), 1717–1727 | MR

[14] A. M. Savchuk, A. A. Shkalikov, “Spektralnye svoistva kompleksnogo operatora Eiri na poluosi”, Funkts. analiz i ego pril., 51:1 (2017), 82–98 | DOI | DOI | MR

[15] B. S. Mityagin, P. Siegl, “Root system of singular perturbations of the harmonic oscillator type operators”, Lett. Math. Phys., 106:2 (2016), 147–167 | DOI | MR | Zbl

[16] E. L. Korotyaev, “Resonances for 1d Stark operators”, J. Spectr. Theory, 7:3 (2017), 699–732, arXiv: 1703.10820 | DOI | MR

[17] E. L. Korotyaev, “O rasseyanii vo vneshnem odnorodnom periodicheskom po vremeni magnitnom pole”, Matem. sb., 180:4 (1989), 491–512 | DOI | MR | Zbl

[18] M. Abramovich, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979 | MR | MR | Zbl

[19] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 2, Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1974 | MR

[20] E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, v. 1, IL, M., 1960 | MR | MR | Zbl

[21] N. E. Firsova, “Pryamaya i obratnaya zadachi rasseyaniya dlya vozmuschennogo operatora Khilla”, Matem. sb., 130:3 (1986), 349–385 | DOI | MR | Zbl