Study of differential operator with summable potential and discontinuous weight function

S. I. Mitrokhin

Ufa mathematical journal, Tome 9 (2017) no. 4, pp. 72-84

Résumé

In the work we propose a new approach for studying differential operators with a discontinuous weight function. We study the spectral properties of a differential operator on a finite segment with separated boundary conditions and with “matching” condition at the discontinuity point of the weight function. We assume that the potential of the operator is a summable function on the segment, on which the operator is considered. For large value of the spectral parameter we obtain an asymptotics for the fundamental system of solutions of the corresponding differential equation. By means of this asymptotics we study the “matching” conditions of the considered differential operator. Then we study the boundary conditions of the considered operator. As a result, we obtain an equation for the eigenvalues of the operator, which an entire function. We study the indicator diagram of the equation for the eigenvalues; this diagram is a regular octagon. In various sectors of the indicator diagram we find the asymptotics for the eigenvalues of the studied differential operator.

Détail

Keywords: spectral theory of differential operators, spectral parameter, summable potential, discontinuous weight function, indicator diagram, asymptotics of eigenvalues.

Bibliographie

[1] Ilin V. A., “O skhodimosti razlozhenii po sobstvennym funktsiyam v tochkakh razryva koeffitsientov differentsialnogo operatora”, Matematicheskie zametki, 22:5 (1977), 698–723

[2] Gekhtman M. M., Zagirov Yu. M., “O maksimalno vozmozhnoi skorosti rosta normirovannykh sobstvennykh funktsii odnogo klassa operatorov tipa Shturma–Liuvillya c nepreryvnoi polozhitelnoi vesovoi funktsiei”, Funktsionalnyi analiz i ego prilozheniya, 27:2 (1993), 85–86 | MR | Zbl

[3] Gekhtman M. M., Aigunov G. A., “K voprosu ob otsenke normirovannykh sobstvennykh funktsii operatora Shturma–Liuvillya s polozhitelnoi vesovoi funktsiei na konechnom otrezke”, UMN, 50:4(304) (1995), 157–158 | MR | Zbl

[4] Mitrokhin S. I., “O spektralnykh svoistvakh differentsialnykh operatorov s razryvnymi koeffitsientami”, Differentsialnye uravneniya, 28:3 (1992), 530–532 | MR | Zbl

[5] Mitrokhin S. I., “O nekotorykh spektralnykh svoistvakh differentsialnykh operatorov vtorogo poryadka s razryvnoi vesovoi funktsiei”, Doklady RAN, 356:1 (1997), 13–15 | MR | Zbl

[6] Khromov A. P., “Operator differentsirovaniya s razryvnoi vesovoi funktsiei”, Mekhanika. Matematika, Sb. nauchnykh trudov, Izd-vo Sarat. un-ta, Saratov, 2009, 88–91

[7] Kuptsov N. P., “Ob analoge teoremy Dirikhle dlya razlozhenii po sobstvennym funktsiyam differentsialnogo uravneniya s razryvnymi koeffitsientami”, Issledovaniya po sovremennym problemam konstruktivnoi teorii funktsii, Fizmatgiz, M., 1961, 201–205

[8] Mukhtarov O. Sh., Kadakal M., “Spektralnye svoistva odnoi zadachi tipa Shturma—Liuvillya s razryvnym vesom”, Sibirskii matematicheskii zhurnal, 46:4 (2005), 860–875 | MR | Zbl

[9] Gurevich A. P., Khromov A. P., “Operatory differentsirovaniya pervogo i vtorogo poryadkov so znakoperemennoi vesovoi funktsiei”, Matematicheskie zametki, 58:1 (1994), 3–15 | MR

[10] Vinokurov V. A., Sadovnichii V. A., “Asimptotika lyubogo poryadka sobstvennykh znachenii i sobstvennykh funktsii kraevoi zadachi Shturma–Liuvillya na otrezke s summiruemym potentsialom”, Differentsialnye uravneniya, 34:10 (1998), 1423–1426 | MR | Zbl

[11] Vinokurov V. A., Sadovnichii V. A., “Asimptotika lyubogo poryadka sobstvennykh znachenii i sobstvennykh funktsii kraevoi zadachi Shturma–Liuvillya na otrezke s summiruemym potentsialom”, Izvestiya RAN. Ser.: matem., 64:4 (2000), 47–108 | DOI | MR | Zbl

[12] Mitrokhin S. I., “O spektralnykh svoistvakh differentsialnogo operatora chetvertogo poryadka s summiruemymi koeffitsientami”, Trudy MIAN, 270, 2010, 188–197 | MR | Zbl

[13] Mitrokhin S. I., “Spektralnye svoistva kraevykh zadach dlya funktsionalno-differentsialnykh uravnenii s integriruemymi koeffitsientami”, Differentsialnye uravneniya, 46:8 (2010), 1085–1093 | MR | Zbl

[14] Mitrokhin S. I., “O spektralnykh svoistvakh odnogo differentsialnogo operatora s summiruemymi koeffitsientami s zapazdyvayuschim argumentom”, Ufimskii matematicheskii zhurnal, 3:4 (2011), 95–115 | MR | Zbl

[15] Savchuk A. M., “Regulyarizovannyi sled pervogo poryadka operatora Shturma–Liuvillya s $\delta$-potentsialom”, UMN, 55:6(336) (2000), 155–156 | DOI | MR | Zbl

[16] Savchuk A. M., Shkalikov A. A., “Operatory Shturma–Liuvillya s singulyarnymi potentsialami”, Matematicheskie zametki, 66:6 (1999), 897–912 | DOI | Zbl

[17] Mitrokhin S. I., “O spektralnykh svoistvakh differentsialnogo operatora s summiruemym potentsialom i gladkoi vesovoi funktsiei”, Vestnik SamGU. Estestvennonauch. seriya., 2008, no. 8(1/67), 172–187

[18] Mitrokhin S. I., Spektralnaya teoriya operatorov: gladkie, razryvnye, summiruemye koeffitsienty, Intuit, M., 2009, 264 pp.

[19] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1977 | MR

[20] Naimark N. A., Lineinye differentsialnye operatory, Nauka, M., 1969, 528 pp. | MR

[21] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu, Nauka, M., 1970, 672 pp. | MR

[22] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983, 352 pp. | MR

[23] Bellman R., Kuk K. L., Differentsialno-raznostnye uravneniya, Mir, M., 1967, 548 pp. | MR

[24] Mitrokhin S. I., “Asimptotika sobstvennykh znachenii differentsialnogo operatora vosmogo poryadka s summiruemym potentsialom s razryvnoi vesovoi funktsiei”, Vtoraya mezhdunarodnaya konferentsiya «Matematicheskaya fizika i ee prilozheniya», Materialy Mezhd. konf. (Samara, 2010), 233–235