Geodesic
The inverse problem of quantum mechanics for a linear potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 56 (1983) no. 1, pp. 74-79 
V. B. Gostev; V. S. Mineev; A. R. Frenkin. The inverse problem of quantum mechanics for a linear potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 56 (1983) no. 1, pp. 74-79. http://geodesic.mathdoc.fr/item/TMF_1983_56_1_a6/
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     title = {The inverse problem of quantum mechanics for a~linear potential},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The applicability of the Gel'fand–Levitan method for solving the inverse problem in the case of potentials that increase unboundedly at infinity is demonstrated for the example of a linear potential. The following cases are considered: 1) change in the normalization of one of the eigenvalues; 2) complete elimination of one of the eigenstates; 3) inclusion in the spectrum of a new state with arbitrary energy. For all three cases, the asymptotic behavior of the new wave functions and the corrections to the reference (linear) potential are calculated.

[1] Eichten E., Gottfried K., Kinoshita T., Lane K. D., Yan T. M., Phys. Rev., D17:11 (1978), 3090-3-117; D21:1 (1980), 203–233

[2] Buchmüller W., Tye S. H. H., The quark-antiquark potential and QCD, Preprint Fermilab-Conf-81/38-THY, Batavia, USA, 1981

[3] Gostev V. B., Mineev V. S., Frenkin A. R., Ekvidistantnyi spektr v kvantovoi mekhanike. Obratnaya zadacha, Preprint 1981/19, Fizicheskii fakultet MGU, M., 1981

[4] Gostev V. B., Mineev V. S., Frenkin A. R., DAN SSSR, 262:6 (1982), 1364–1367 | MR

[5] Weil H., Math. Ann., 68 (1910), 220–269 | DOI | MR

[6] Titchmarsh E. Ch., Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, t. I, gl. 2, 4 ; т. II, гл. 19, ИЛ, М., 1960 | MR | Zbl

[7] Spravochnik po spetsialnym funktsiyam, eds. Abramovits M., Stigan I. M., Nauka, M., 1979, 264–270 | MR

[8] Nyuton R., Teoriya rasseyaniya voln i chastits, Mir, M., 1969, 560–563 | MR

[9] Abraham P. B., Moses H. E., Phys. Rev., A22:4 (1980), 1333–1340 | DOI | MR

[10] Adamyan M. N., TMF, 48:1 (1981), 70–79 | MR

[11] Shadan K., Sabate P., Obratnye zadachi v kvantovoi teorii rasseyaniya, Mir, M., 1980, 408 pp. | MR

[12] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966, 521–525 | MR | Zbl