Geodesic
A perturbed boundary eigenvalue problem for the Schrödinger operator on an interval
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 4, pp. 679-698 Cet article a éte moissonné depuis la source Math-Net.Ru

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A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential $\mu^{-1}V((x-x_0)\varepsilon^{-1})$, where $0<\varepsilon\ll1$ and $\mu$ is an arbitrary parameter such that there exists $\delta>0$ for which $\varepsilon/\mu=o(\varepsilon^\delta)$. It is shown that the eigenvalues of this operator converge, as $\varepsilon\to0$, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed. 
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     title = {A perturbed boundary eigenvalue problem for the {Schr\"odinger} operator on an interval},
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I. Kh. Khusnullin. A perturbed boundary eigenvalue problem for the Schrödinger operator on an interval. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 4, pp. 679-698. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_4_a6/

[1] Landau L. D., Livshits E. M., Teoreticheskaya fizika, v. 3, Kvantovaya mekhanika. Nerelyativistskaya teoriya, Nauka, M., 1974

[2] Simon B., “The bound state of weakly coupled Schrödinger operators in one and two dimensions”, Ann. Phys., 97 (1976), 279–288 | DOI | MR | Zbl

[3] Klaus M., “On the bound state of Schrödinger operators in one dimension”, Ann. Phys., 108 (1977), 288–300 | DOI | MR | Zbl

[4] Blankenbecler R., Goldberger M. L., Simon B., “The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians”, Ann. Phys., 108 (1977), 69–78 | DOI | MR

[5] Klaus M., Simon B., “Coupling constant thresholds in nonrelativistic quantum mechanics. I: Short-range two-body case”, Ann. Phys., 130 (1980), 251–281 | DOI | MR | Zbl

[6] Gadylshin R. R., “O lokalnykh vozmuscheniyakh operatora Shredingera na osi”, Teor. i matem. fiz., 132 (2002), 97–104 | MR

[7] Gadylshin R. R., “O lokalnykh vozmuscheniyakh operatora Shredingera na ploskosti”, Teor. i matem. fiz., 138 (2004), 41–54 | MR

[8] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[9] Bikmetov A. R., “Asimptotiki sobstvennykh elementov kraevykh zadach operatora Shredingera s bolshim potentsialom, lokalizovannym na malom mnozhestve”, Zh. vychisl. matem. i matem. fiz., 2006, no. 4, 666–681 | MR

[10] Bikmetov A. R., Gadylshin R. R., “O spektre operatora Shredingera s rastuschim potentsialom, lokalizovannym na szhimayuschemsya mnozhestve”, Matem. zametki, 79:5 (2006), 787–790 | MR | Zbl

[11] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[12] Bikmetov A. R., Borisov D. I., “O diskretnom spektre operatora Shredingera, vozmuschennogo ogranichennym potentsialom s malym nositelem”, Teor. i matem. fiz., 43:3 (2005), 372–384 | MR

[13] Tikhonov A. N., Differentsialnye uravneniya, Nauka, M., 1980 | MR

[14] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 | MR | Zbl

[15] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[16] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

[17] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1987 | MR