About the directed disturbance of discrete operators

E. M. Maleko

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 8 (2009), pp. 44-60

Résumé

Under the directed perturbed operator we will understand such an operator, which is built from the disturbance of a discrete operator by the operators of shift, in such a way that one of the eigunefunctions of the built operator is any preassigned nonvanishing function from the domain of operator. By solving problems of theoretical function we often use the spectrum of researched operator, and in many cases only its discrete part, but sometimes together with eigen (and attached) functions. Probably for the researcher it is important to know that it is possible by means of disturbing the given operator to receive an operator with resembling on the primary by the spectrum, and preassigned function which is any by number to the eigenfunction of the disturbed operator.

Détail

Keywords: spectrum, the discrete operator, Hilbert space.

Bibliographie

[1] Krein M. G., “Ob odnom obschem metode razlozheniya polozhitelno opredelennykh yader na elementarnye proizvedeniya”, DAN SSSR, 1946, no. 53, 3–6 | MR | Zbl

[2] Krein M. G., “Pro ermitovi operatori z napryamnimi funktsionalami”, Zbirnik prats institutu matematiki AN URSR, 1948, no. 10, 83–105 | MR

[3] V. V. Dubrovskii i dr., “Novyi metod priblizhennogo vychisleniya pervykh sobstvennykh chisel spektralnoi zadachi gidrodinamicheskoi ustoichivosti techeniya Puazeilya v krugloi trube”, DAN, 380:2 (2001), 160–163 | MR

[4] V. V. Dubrovskii i dr., “Novyi metod vychisleniya pervykh sobstvennykh chisel spektralnoi zadachi gidrodinamicheskoi teorii ustoichivosti techeniya vyazkoi zhidkosti mezhdu dvumya vraschayuschimisya tsilindrami”, DAN, 381:3 (2001), 320–324 | MR

[5] Kadchenko S. I., Kravchenko V. F., Dzhiganchina N. S., “Ustoichivost ploskoparallelnogo techeniya Kutta”, Elektromagnitnye volny i elektronnye sistemy, 10:1,2 (2005), 10–21

[6] Kadchenko S. I., Kinzina I. I., “Lineinye uravneniya dlya priblizhennogo vychisleniya sobstvennykh chisel vozmuschennykh samosopryazhennykh operatorov”, Elektromagnitnye volny i elektronnye sistemy, 10:6 (2005), 4–12

[7] Kadchenko S. I., Kinzina I. I., “Vychislenie sobstvennykh znachenii vozmuschennykh diskretnykh poluogranichennykh operatorov”, Vychislitelnaya matematika i matematicheskaya fizika, 46:7 (2006), 1265–1272 | MR

[8] Beitmen G., Vysshie transtsendentnye funktsii: funktsii Besselya, funktsii parobolicheskogo tsilindra, ortogonalnye mnogochleny, izd. 2-e, ster., Nauka, M., 1974, 296 pp.

[9] Maleko E. M., “O vosstanovlenii deistvuyuschikh v separabelnom gilbertovom prostranstve operatorov”, Matematika. Mekhanika. Informatika, tez. dokl. Vseros. nauch. konf., ChelGU, Chelyabinsk, 2006

[10] Maleko E. M., “O predstavlenii operatora s zaranee zadannoi sobstvennoi funktsiei”, Vestnik MaGU. Ser.: Matematika, 2006, no. 9, 186 s.

[11] Naimark M. A., Lineinye differentsialnye operatory, Gostekhteoretizdat, M., 1954, 352 pp.

[12] Sadovnichii V. A., Teoriya operatorov, izd. 2-e, Izd-vo Mosk. un-ta, M., 1986, 386 pp. | MR