Carleson series of resonances in the Regge problem

S. A. Ivanov ; B. S. Pavlov

Izvestiya. Mathematics , Tome 12 (1978) no. 1, pp. 21-51.

Résumé

The scattering problem is investigated for a ystem of differential equations $-y''=k^2Ay$ on the interval $[0,a]$, where $A$ is a positive matrix-valued function which jumps to 1 for $x>a$. The scattering matrix for large spectral parameter is studied, the system of resonances is described, and an expression for the resonance states in terms of the Jost solution is given. A relation is established between the resonances and the poles of the analytic continuation of the Green function. It is proved that the syrtem of resonance states corresponding to complex zeros of the scattering matrix has serial structure; namely, it splits into $n$ Carleson series. The completeness of the system of resonances is investigated, and it is established that this system forms a Riesz basis in the corresponding space with the energy metric. Bibliography: 23 titles.

Bibliographie

[1] Regge T., “Analitic properties of the scattering matrix”, Nuovo Cimento, 8:5 (1958), 671–679 ; Matematika, 7:4 (1963), 83–89 | DOI | MR | Zbl

[2] Laks P., Fillips R., Teoriya rasseyaniya, Mir, M., 1971 | MR

[3] Adamyan V. M., Arov D. Z., “Ob unitarnykh stsepleniyakh poluunitarnykh operatorov”, Matematicheskie issledovaniya, t. I, no. 2, Kishinev, 1966, 3–63 | MR

[4] Pavlov B. S., “Faktorizatsiya matritsy rasseyaniya i seriinaya struktura ee kornei”, Izv. AN SSSR. Ser. matem., 37 (1973), 217–246 | MR | Zbl

[5] Pekker M. A., “O nefizicheskom liste dlya uravneniya struny”, Problemy matematicheskogo analiza, vyp. 5, LGU, L., 1975, 133–152 | MR

[6] Pavlov B. S., “Teoriya rasseyaniya i nefizicheskii list dlya sistem obyknovennykh differentsialnykh uravnenii”, Dokl. AN SSSR, 193 (1970), 36–39 | Zbl

[7] Pavlov B. S., “O polnote nabora rezonansnykh sostoyanii dlya sistemy differentsialnykh uravnenii”, Dokl. AN SSSR, 196:1 (1971), 1272–1275 | MR | Zbl

[8] Nikolskii N. K., Pavlov B. S., “Bazisy iz sobstvennykh vektorov vpolne neunitarnykh szhatii i kharakteristicheskaya funktsiya”, Izv. AN SSSR. Ser. matem., 34 (1970), 90–133 | MR

[9] Potapov V. P., “Multiplikativnaya struktura analiticheskikh nerastyagivayuschikh matrits funktsii”, Tr. Mosk. Matem. Obschestva, 4, 1955, 125–236 | MR | Zbl

[10] S.-Nad B., Foiash Ch., Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970 | MR

[11] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[12] Agranovich Z. S., Marchenko V. A., Obratnaya zadacha teorii rasseyaniya, Kharkovskii universitet, Kharkov, 1960 | Zbl

[13] Danford N., Shvarts Dzh., Lineinye operatory, t. 2, Mir, M., 1966

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

[15] Helson H., Lectures on invariant subspaces, N. Y., London, 1964 | MR | Zbl

[16] Pavlov B. S., “O sovmestnoi polnote sistemy sobstvennykh funktsii szhatiya i ego sopryazhennogo”, Problemy matematicheskoi fiziki, vyp. 5, LGU, 1971, 101–112

[17] Buslaeva M. V., “O kornevykh podprostranstvakh szhatiya, porozhdaemogo vnutrennei funktsiei”, Zapiski nauchnykh seminarov LOMI, 42, 1974, 78–84 | MR | Zbl

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

[19] Feschenko S. F., Shkil N. I., Nikolenko L. D., Asimptoticheskie metody v teorii lineinykh differentsialnykh uravnenii, Naukova dumka, Kiev, 1966 | MR | Zbl

[20] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[21] Kucherenko V. V., “Asimptotika resheniya sistemy $\displaystyle A\left(x, -ih\frac{\partial}{\partial x}\right)u=0$ pri $h\to\infty$ v sluchae kharakteristik peremennoi kratnosti”, Izv. AN SSSR. Ser. matem., 38 (1974), 625–662 | Zbl

[22] Evgrafov M. A., Analiticheskie funktsii, Nauka, M., 1965 | MR | Zbl

[23] Pavlov B. S., “Spektralnyi analiz differentsialnogo operatora s razmazannym granichnym usloviem”, Problemy matematicheskoi fiziki, vyp. 6, LGU, 1973, 101–119