Geodesic
Induced potential and Klauder phenomenon for even confinement potentials with singularity $\lambda x^{-2}$
Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 1, pp. 45-57 
V. B. Gostev; V. S. Mineev; A. R. Frenkin. Induced potential and Klauder phenomenon for even confinement potentials with singularity $\lambda x^{-2}$. Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 1, pp. 45-57. http://geodesic.mathdoc.fr/item/TMF_1986_68_1_a3/
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     author = {V. B. Gostev and V. S. Mineev and A. R. Frenkin},
     title = {Induced potential and {Klauder} phenomenon for even confinement potentials with singularity $\lambda x^{-2}$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     url = {http://geodesic.mathdoc.fr/item/TMF_1986_68_1_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For the example of a one-dimensional oscillator with singular perturbation the occurrence of the strongly singular potential $A(\lambda)\delta(x)|x|^{-1}$ induced by this perturbation is investigated. The appearance of such a potential can be regarded as a generalization of the Klauder phenomenon ($A=\operatorname{const}$). The existence of induced potentials leads to two physically acceptable sets of even states of a perturbed oscillator, but only one of these goes over continuously into the set of even states of the harmonic oscillator when the perturbation is switched off.

[1] Klauder J., Acta Phys. Austriaca Suppl., 11 (1973), 341–387

[2] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, T. 4, Mir, M., 1982, s. 77. | MR

[3] Ezawa H., Klauder J., Shepp L., J. Math. Phys., 16 (1975), 783–799 | DOI | MR

[4] Landau L. D., Lifshits E. M., Kvantovaya mekhanika, Nauka, M., 1974 | MR

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

[6] Baz A. I., Zeldovich Ya. B., Perelomov A. M., Rasseyanie, reaktsii i raspady v nerelyativistskoi kvantovoi mekhanike, Nauka, M., 1971, 513–529. | Zbl

[7] Mors F. M., Feshbakh G., Metody teoreticheskoi fiziki, t. II, IL, M., 1960, 595–596. | MR

[8] Calogero F., J. Math. Phys., 10 (1969), 2191–2220 | DOI | MR

[9] Malkin I. A., Manko V. I., Dinamicheskie simmetrii i kogerentnye sostoyaniya kvantovykh sistem, Nauka, M., 1979, 99–109. | MR

[10] Fon Neiman Dzh., Matematicheskie osnovy kvantovoi mekhaniki, Nauka, M., 1964 | MR

[11] Andreev I. V., Khromodinamika i zhestkie protsessy pri vysokikh energiyakh, Nauka, M., 1981, 123–124. | Zbl

[12] Abramovits M., Stigan I. (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979, 321–325. | MR

[13] Flyugge Z., Zadachi po kvantovoi mekhanike, t. 1, Mir, M., 1974, 53–54.