On higher-order differential operators with a regular singularity

V. A. Yurko

Sbornik. Mathematics, Tome 186 (1995) no. 6, pp. 901-928

Résumé

A boundary-value problem for the non-self-adjoint differential operators $$ \ell y\equiv y^{(n)}+\sum_{j=0}^{n-2}\biggl(\frac{\nu_j}{x^{n-j}}+q_j(x)\biggr)y^{(j)}, \qquad 0<x<T, $$ with a regular singularity at zero is investigated. Theorems are obtained on completeness, on the expansion with respect to the eigen- and associated functions of the boundary-value problem on a finite interval, and on equiconvergence. In addition, the inverse problem is investigated.

Bibliographie

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

[2] Kostyuchenko A. G., Sargsyan I. S., Raspredelenie sobstvennykh znachenii. Samosopryazhennye obyknovennye differentsialnye operatory, Nauka, M., 1979 | MR | Zbl

[3] Birkhoff G. D., “Boundary value and expansion problems of ordinary linear differential equations”, Trans. Amer. Maths. Soc., 9:4 (1908), 373–397 | DOI | MR

[4] Tamarkin Ya. D., O nekotorykh obschikh zadachakh teorii obyknovennykh lineinykh differentsialnykh uravnenii, Petrograd, 1917 | Zbl

[5] Stone M. H., “A comparison of the series of Fourier and Birkhoff”, Trans. Amer. Math. Soc., 28:4 (1926), 695–761 | DOI | MR

[6] Khromov A. P., “Razlozhenie po sobstvennym funktsiyam obyknovennykh lineinykh differentsialnykh operatorov s neregulyarnymi raspadayuschimisya kraevymi usloviyami”, Matem. sb., 70 (112):3 (1966), 310–329 | MR

[7] Sakhnovich L. A., “Obratnaya zadacha dlya differentsialnykh operatorov poryadka $n>2$ s analiticheskimi koeffitsientami”, Matem. sb., 46 (88):1 (1958), 61–76 | MR

[8] Leibenzon Z. L., “Obratnaya zadacha spektralnogo analiza dlya differentsialnykh operatorov vysshikh poryadkov”, Tr. MMO, 15, URSS, M., 1966, 70–144 | MR | Zbl

[9] Beals R., “The inverse problem for ordinary differential operators on the line”, Amer. J. Math., 107:2 (1985), 281–366 | DOI | MR | Zbl

[10] Khachatryan I. G., “O nekotorykh obratnykh zadachakh dlya differentsialnykh operatorov vysshikh poryadkov na poluosi”, Funktsion. analiz i ego pril., 17:1 (1983), 40–52 | MR | Zbl

[11] Yurko V. A., “Obratnaya zadacha dlya differentsialnykh uravnenii s osobennostyu”, Differents. uravneniya, 28:8 (1992), 1355–1362 | MR | Zbl

[12] Vazov V., Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1968

[13] Vatson G., Teoriya besselevykh funktsii, T. 1, Gostekhizdat, M.–L., 1949

[14] Titchmarsh E., Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, T. 1, IL, M., 1960

[15] Gasymov M. G., “Razlozhenie po sobstvennym funktsiyam nesamosopryazhennogo differentsialnogo operatora vtorogo poryadka s osobennostyu v nule”, Trudy letnei shkoly po spektralnoi teorii operatorov i teorii predstavleniya grupp, Izd-vo ELM, Baku, 1975 | MR | Zbl

[16] Stashevskaya V. V., “Ob obratnykh zadachakh spektralnogo analiza dlya odnogo klassa differentsialnykh uravnenii”, DAN SSSR, 93:3 (1953), 409–412

[17] Yurko V. A., “On higher order differential operators with a singular point”, Inverse Problems, 9:4 (1993), 495–502 | DOI | MR | Zbl

[18] Yurko V. A., “Vosstanovlenie nesamosopryazhennykh differentsialnykh operatorov na poluosi po matritse Veilya”, Matem. sb., 182:3 (1991), 431–456

[19] Yurko V. A., “Vosstanovlenie differentsialnykh operatorov vysshikh poryadkov”, Differents. uravneniya, 25:9 (1989), 1540–1550 | MR | Zbl

[20] Yurko V. A., “Solution of the Boussinesq equation on the half-line by the inverse problem method”, Inverse Problems, 7 (1991), 727–738 | DOI | MR

[21] Privalov I. I., Vvedenie v teoriyu funktsii kompleksnogo peremennogo, Nauka, M., 1967 | MR

[22] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR