Spectral properties of one infinite tridiagonal matrix

G. V. Garkavenko ; N. B. Uskova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 1, Tome 199 (2021), pp. 31-42.

Résumé

Using the method of similar operators, we obtain conditions under which an infinite tridiagonal matrix can be transformed to the diagonal (block-diagonal) form by a similarity transformation. Asymptotic estimates of the eigenvalues and eigenvectors are also obtained.

Détail

Mots-clés : infinite tridiagonal matrices
Keywords: eigenvalue, eigenvector, method of similar operators.

Bibliographie

[1] Baskakov A. G., “Metody abstraktnogo garmonicheskogo analiza v teorii vozmuschenii lineinykh operatorov”, Sib. mat. zh., 24 (1983), 21–39

[2] Baskakov A. G., “Spektralnyi analiz vozmuschennykh nekvazianaliticheskikh i spekralnykh operatorov”, Izv. RAN. Ser. mat., 58:4 (1994), 3–32 | Zbl

[3] Baskakov A. G., Derbushev A. V., Scherbakov A. O., “Metod podobnykh operatorov v spektralnom analize operatora Diraka s negladkim potentsialom”, Izv. RAN. Ser. mat., 75:3 (2011), 3–28 | MR | Zbl

[4] Baskakov A. G., Krishtal I. A., Uskova N. B., “Metod podobnykh operatorov v issledovanii spektralnykh svoistv vozmuschennykh differentsialnykh operatorov pervogo poryadka”, Itogi nauki tekhn. Ser. Sovr. mat. prilozh. Temat. obzory., 171 (2019), 3–18

[5] Baskakov A. G., Polyakov D. M., “Metod podobnykh operatorov v spektralnom analize operatora Khilla s negladkim potentsialom”, Mat. sb., 208:1 (2017), 3–47 | MR | Zbl

[6] Baskakov A. G., Uskova N. B., “Obobschennyi metod Fure dlya sistemy differentsialnykh uravnenii pervogo poryadka i gruppy operatorov”, Differ. uravn., 54:2 (2018), 276–280 | Zbl

[7] Baskakov A. G., Uskova N. B., “Spektralnyi analiz differentsialnykh operatorov s involyutsiei i gruppy operatorov”, Differ. uravn., 54:9 (2018), 1287–1291 | Zbl

[8] Baskakov A. G., Uskova N. B., “Metod Fure dlya differentsialnykh uravnenii pervogo poryadka i gruppy operatorov”, Ufim. mat. zh., 10:3 (2018), 11–34 | MR | Zbl

[9] Broitigam I. N., Polyakov D. M., “Asimptotika sobstvennykh znachenii beskonechnykh blochnykh matrits”, Ufim. mat. zh., 11:3 (2019), 10–29 | Zbl

[10] Burlutskaya M. Sh., “Klassicheskoe i obobschennoe reshenie smeshannoi zadachi dlya sistemy uravnenii pervogo poryadka s nepreryvnym potentsialom”, Zh. vychisl. mat. mat. fiz., 59:3 (2019), 380–390 | MR | Zbl

[11] Burlutskaya M. Sh., “Smeshannaya zadacha dlya sistemy differentsialnykh uravnenii pervogo poryadka s nepreryvnym potentsialom”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 16:2 (2016), 145–151 | MR | Zbl

[12] Burlutskaya M. Sh., Khromov A. P., “Operator Diraka s potentsialom spetsialnogo vida i periodicheskimi kraevymi usloviyami”, Differ. uravn., 54:5 (2018), 592–601 | MR | Zbl

[13] Burlutskaya M. Sh., Khromov A. P., “Funktsionalno-differentsialnye operatory s involyutsiei i operatory Diraka s periodicheskimi kraevymi usloviyami”, Dokl. RAN., 454:1 (2014), 15–17 | MR | Zbl

[14] Gantmakher F. R., Teoriya matrits, Nauka, M., 1967

[15] Garkavenko G. V., “O diagonalizatsii nekotorykh klassov lineinykh operatorov”, Izv. vuzov. Mat., 1994, no. 11, 14–19 | MR | Zbl

[16] Garkavenko G. V., Uskova N. B., “Spektralnyi analiz odnogo klassa raznostnykh operatorov s rastuschim potentsialom”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 16:4 (2016), 395–402 | MR | Zbl

[17] Garkavenko G. V., Uskova N. B., “Asimptotika sobstvennykh znachenii raznostnogo operatora s rastuschim potentsialom i polugruppy operatorov”, Mat. fiz. kompyut. model., 20:4 (2017), 6–17 | MR

[18] Garkavenko G. V., Uskova N. B., “Spektralnyi analiz raznostnykh operatorov vtorogo poryadka s rastuschim potentsialom”, Tavrich. vestn. inform. mat., 2015, no. 3 (28), 40–48

[19] Garkavenko G. V., Uskova N. B., “Ob odnom modelnom primere metoda podobnykh operatorov”, Mat. 20 Mezhdunar. Saratov. zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya» (Saratov, 28 yanvarya — 1 fevralya 2020 g.), Nauchnaya kniga, Saratov, 2020, 112–115

[20] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965

[21] Danford N., Shvarts Dzh. T., Lineinye operatory. Spektralnye operatory. T. III, Mir, M., 1974

[22] Katrakhov V. V., Sitnik S. M., “Metod operatorov preobrazovaniya i kraevye zadachi dlya singulyarnykh ellipticheskikh uravnenii”, Sovr. mat. Fundam. napravl., 64:2 (2018), 211–426 | MR

[23] Krishtal I. A., Uskova N. B., “Spektralnye svoistva differentsialnykh operatorov pervogo poryadka s involyutsiei i gruppy operatorov”, Sib. elektron. mat. izv., 16. (2019), 1091–1132 | MR | Zbl

[24] Kuk R., Beskonechnye matritsy i prostranstva posledovatelnostei, GIFML, M., 1960

[25] Mak-Kinsi Dzh., Vvedenie v teoriyu igr, GIFML, M., 1960

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

[27] Rudin U., Funktsionalnyi analiz, Mir, M., 1975

[28] Sitnik S. M., Shishkina E. L., Metod operatorov preobrazovaniya dlya differentsialnykh uravnenii s operatorami Besselya, Fizmatlit, M., 2019

[29] Skrynnikov A. V., “O kvazinilpotentnom variante metoda Fridrikhsa v teorii podobiya lineinykh operatorov”, Funkts. anal. prilozh., 17:3 (1983), 89–90 | MR | Zbl

[30] Uskova N. B., “Matrichnyi analiz spektralnykh proektorov vozmuschennykh samosopryazhennykh operatorov”, Sib. elektron. mat. izv., 16 (2019), 369–405 | MR | Zbl

[31] Uskova N. B., “Spektralnye svoistva operatora Diraka s negladkim potentsialom obschego vida i gruppy operatorov”, Differ. uravn., 55:8 (2019), 1154–1158 | Zbl

[32] Uskova N. B., Garkavenko G. V., “Teorema o rasscheplenii lineinykh operatorov i asimptotika sobstvennykh znachenii raznostnykh operatorov s rastuschim potentsialom”, Sib. zh. chist. prikl. mat., 18:1 (2018), 91–106 | Zbl

[33] Baskakov A. G., Krishtal I. A., Uskova N. B., “Linear differential operator with an involution as a generator of an operator group”, J. Oper. Matr., 12:3 (2018), 723–756 | DOI | Zbl

[34] Baskakov A. G., Krishtal I. A., Uskova N. B., “Similarity techniques in the spectral analysis of perturbed operator matrices”, J. Math. Anal. Appl., 477 (2019), 930–960 | DOI | MR | Zbl

[35] Garkavenko G. V., Zgolich A. R., Uskova N. B., “Spectral analysis of a difference operator with a growing potential”, J. Phys. Conf. Ser., 973 (2018), 012053 | DOI

[36] Garkavenko G. V., Zgolich A. R., Uskova N. B., “Spectral analysis of one class of the integro-differential operators”, J. Phys. Conf. Ser., 1203 (2019), 012102 | DOI

[37] Hinkkannen A., “On the diagonalization of a certain class of operators”, Michigan Math. J., 32:3 (1985), 349–359 | DOI | MR

[38] Ikebe Y., Asai N., Miyazaki Y., Cai D., “The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application”, Linear Algebra Appl., 241–243 (1996), 599–618 | DOI | MR | Zbl

[39] Janas J., Naboko S., “Infinite Jacobi matrices with unbounded entries: Asymptotics of eigenvalues and the transformation operator approach”, SIAM J. Math. Anal., 36:2 (2004), 643–658 | DOI | MR | Zbl

[40] Malejki M., “Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices”, Opuscula Math., 30:3 (2010), 311–330 | DOI | MR | Zbl

[41] Malejki M., “Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices”, Linear Algebra Appl., 431 (2009), 1952–1970 | DOI | MR | Zbl

[42] Malejki M., “Eigenvalues for some complex infinite tridiagonal matrices”, J. Adv. Math. Comp. Sci., 26:5 (2018), 1–9 | DOI