Schr\"odinger Operators with Singular Potentials

M. I. Neiman-Zade ; A. M. Savchuk

Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 262-271.

Résumé

Schrödinger operators are investigated whose potentials represent generalized functions. A problem of physically correct determination of such operators is solved by two different methods in the case of one and several variables. In the first case, the potential must represent an element of the negative space $W^{-1}_2$; then, the operator obtained can be analyzed in greater detail. In the second case, a requirement is imposed on the potential that it must belong to the class of multipliers from $W^{1}_2$ to $W^{-1}_2$. In addition, the space of multipliers is analyzed.

Bibliographie

[1] Landau L. D., Lifshits E. M., Kvantovaya mekhanika. Nerelyativistskaya teoriya, T. 3, Teoreticheskaya fizika, Nauka, M., 1989

[2] Berezin F. A., Faddeev L. D., “Zamechaniya ob uravnenii Shredingera s singulyarnym potentsialom”, DAN SSSR, 137:7 (1961), 1011–1014 | MR | Zbl

[3] Minlos R. A., Faddeev L. D., “O tochechnom vzaimodeistvii dlya sistem iz trekh chastits v kvantovoi mekhanike”, DAN SSSR, 141:6 (1961), 1335–1338 | MR

[4] Berezin F. A., “O modeli Li”, Mat. sb., 60:4 (1963), 425–446 | MR

[5] Gesztezy F., Simon B., “Rank-one perturbations at infinite coupling”, J. Funct. Anal., 128 (1995), 245–252 | DOI | MR

[6] Kiselev A., Simon B., “Rank one perturbations with infinitesimal coupling”, J. Funct. Anal., 130 (1995), 345–356 | DOI | MR | Zbl

[7] Koshmanenko V., Karwowski W., Ota S., “Schrödinger operator perturbed by operators related to null sets”, Positivity, 2:1 (1998), 77–99 | DOI | MR | Zbl

[8] Fragela A. K., “O vozmuschenii poligarmonicheskogo operatora potentsialami s malymi nositelyami”, DAN SSSR, 245:1 (1979), 34–36 | MR | Zbl

[9] Shondin Yu. G., “Vozmuscheniya na tonkikh mnozhestvakh vysokoi korazmernosti ellipticheskikh operatorov i teoriya rasshirenii v prostranstve s indefinitnoi metrikoi”, Zap. nauch. seminarov POMI, 222, 1995, 246–292 | MR

[10] Gunson J., “Perturbation theory for a Sturm–Liouville problem with an interior singularity”, Proc. Roy. Soc. London A, 414 (1987), 255–269 | DOI | MR | Zbl

[11] Kurasov P., “On the Coulomb potentials in one dimension”, J. Phys. A, 29:8 (1996), 1767–1771 | DOI | MR | Zbl

[12] Albeverio S., Gesztezy F., Hoegh-Krohn R., Holden H., Solvable models in quantum mechanics, Springer, Berlin etc., 1988 | MR | Zbl

[13] Koshmanenko V. D., “Vozmuscheniya samosopryazhennykh operatorov singulyarnymi bilineinymi formami”, Ukr. mat. zhurn., 41:1 (1989), 3–18 | MR

[14] Mazya V. G., Shaposhnikova T. A., Multiplikatory v prostranstvakh differentsiruemykh funktsii, LGU, L., 1986 | MR

[15] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu, Nauka, M., 1970 | MR | Zbl

[16] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[17] Ince E. L., Ordinary differential equations, Dover, New York, 1956 | MR

[18] Zhikov V. V., “Ob obratnykh zadachakh Shturma–Liuvillya na konechnom otrezke”, Izv. AN SSSR. Ser. mat., 31:5 (1967), 965–976 | Zbl

[19] Seba P., “Some remarks on the $\delta'$-interaction in one dimension”, Repts. Math. Phys., 24:1 (1986), 111–120 | DOI | MR | Zbl

[20] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, T. 2, Mir, M., 1978 | MR

[21] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1959

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