Asymptotics of the spectrum of a periodic boundary value problem for a differential operator with a summable potential

S. I. Mitrokhin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 1, pp. 136-149

Résumé

The spectrum of a differential operator of a high odd order with summable potential is studied. The boundary conditions are periodic. The differential equation that defines the differential operator is reduced to the Volterra integral equation. Solving this equation by Picard's method of successive approximations, we find the asymptotics of the fundamental system of solutions of the original differential equation. This fundamental system of solutions is used for the study of periodic boundary conditions. As a result, an equation for the eigenvalues of the differential operator is derived. This equation is a determinant of high order, which is an entire function of the spectral parameter. The indicator diagram corresponding to this function is investigated. The indicator diagram is a regular polygon and determines the location of the eigenvalues of the operator under consideration. As a result, the asymptotic behavior of the eigenvalues of the operator is found in each of the sectors of the complex plane determined by the indicator diagram (of 15th order).

Détail

Keywords: spectral parameter, differential operator, summable potential, periodic boundary conditions, asymptotics of solutions of a differential equation, asymptotics of the spectrum.

Bibliographie

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

[2] Marchenko V.A., “Nekotorye voprosy teorii odnomernykh lineinykh differentsialnykh operatorov vtorogo poryadka”, Tr. Mosk. mat. obschestva, 1952, 327–420

[3] Lidskii V.B., Sadovnichii V.A., “Asimptoticheskie formuly dlya kornei odnogo klassa tselykh funktsii”, Mat. sb., 65:4 (1968), 558–566

[4] Lidskii V.V., Sadovnichii V.A., “Regulyarizovannye summy kornei odnogo klassa tselykh funktsii”, Funktsionalnyi analiz i ego prilozheniya, 1:2 (1967), 52–59 | MR | Zbl

[5] Sadovnichii V.A., “O sledakh obyknovennykh differentsialnykh operatorov vysshikh poryadkov”, Mat. sb., 72:2 (1967), 293–310

[6] Babadzhanov B.A., Khasanov A.B., Yakhshimuratov A.B., “Ob obratnoi zadache dlya kvadratichnogo puchka operatorov Shturma - Liuvillya s periodicheskim potentsialom”, Differents. uravneniya, 41:3 (2005), 298–305 | MR | Zbl

[7] Fedotov A.A., Schetka E.V., “Kompleksnyi metod VKB dlya raznostnogo uravneniya Shredingera, potentsial kotorogo - trigonometricheskii polinom”, Algebra i analiz, 29:2 (2017), 193–219

[8] Mitrokhin S.I., “O formulakh regulyarizovannykh sledov dlya differentsialnykh operatorov vtorogo poryadka s razryvnymi koeffitsientami”, Vestn. MGU. Ser. Matematika. Mekhanika, 1986, no. 6, 3–6 | MR

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

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

[11] Abdullaev A.R., Skachkova E.A., “Periodicheskaya kraevaya zadacha dlya differentsialnogo uravneniya chetvertogo poryadka”, Izv. vuzov. Matematika, 2013, no. 12, 3–10 | Zbl

[12] Badanin A.V., Korotyaev E.L., “Spektralnye otsenki dlya periodicheskogo operatora chetvertogo poryadka”, Algebra i analiz, 22:5 (2010), 1–48 | MR

[13] Polyakov D.M., “Spektralnyi analiz differentsialnogo operatora chetvertogo poryadka s periodicheskimi i antiperiodicheskimi kraevymi usloviyami”, Algebra i analiz, 27:5 (2015), 117–152

[14] Polyakov D.M., “O spektralnykh svoistvakh differentsialnogo operatora chetvertogo poryadka s periodicheskimi i antiperiodicheskimi kraevymi usloviyami”, Izv. vuzov. Matematika, 2015, no. 5, 75–79 | Zbl

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

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

[17] Mitrokhin S.I., “O spektralnykh svoistvakh differentsialnykh operatorov nechetnogo poryadka s summiruemym potentsialom”, Differents. uravneniya, 47:2 (2011), 1808–1811 | MR | Zbl

[18] Mitrokhin S.I., “O spektralnykh svoistvakh semeistva differentsialnykh operatorov vysokogo chetnogo poryadka s summiruemym potentsialom”, Vest. Volgograd. gos. un-ta. Ser. Fizika. Matematika, 2016, no. 4, 121–135 | MR

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

[20] Mitrokhin S.I., “Periodicheskaya kraevaya zadacha dlya differentsialnogo operatora chetvertogo poryadka s summiruemym potentsialom”, Vladikavkaz. mat. zhurn., 19:4 (2017), 35–49 | MR

[21] Naimark M.A., Lineinye differentsialnye operatory, Nauka, M., 1969, 528 pp.

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