Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces

I. V. Sadovnichaya

Informatics and Automation, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 296-324.

Résumé

The problem of equiconvergence of spectral decompositions corresponding to the systems of root functions of two one-dimensional Dirac operators $\mathcal L_{P,U}$ and $\mathcal L_{0,U}$ with potential $P$ summable on a finite interval and Birkhoff-regular boundary conditions is analyzed. It is proved that in the case of $P\in L_\varkappa[0,\pi]$, $\varkappa\in(1,\infty]$, equiconvergence holds for every function $\mathbf f\in L_\mu[0,\pi]$, $\mu\in[1,\infty]$, in the norm of the space $L_\nu[0,\pi]$, $\nu\in[1,\infty]$, if the indices $\varkappa,\mu$, and $\nu$ satisfy the inequality $1/\varkappa+1/\mu-1/\nu\le1$ (except for the case when $\varkappa=\nu=\infty$ and $\mu=1$). In particular, in the case of a square summable potential, the uniform equiconvergence on the interval $[0,\pi]$ is proved for an arbitrary function $\mathbf f\in L_2[0,\pi]$.

Bibliographie

[1] Albeverio S., Hryniv R., Mykytyuk Ya., “Inverse spectral problems for Dirac operators with summable potentials”, Russ. J. Math. Phys., 12:4 (2005), 406–423 | MR | Zbl

[2] Amirov R.Kh., Guseinov I.M., “Nekotorye klassy operatorov Diraka s singulyarnymi potentsialami”, Dif. uravneniya, 40:7 (2004), 999–1001 | MR | Zbl

[3] Berg I., Lëfstrëm I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[4] Birkhoff G.D., “On the asymptotic character of the solutions of certain linear differential equations containing a parameter”, Trans. Amer. Math. Soc., 9:2 (1908), 219–231 | DOI | MR

[5] Birkhoff G.D., “Boundary value and expansion problems of ordinary linear differential equations”, Trans. Amer. Math. Soc., 9:4 (1908), 373–395 | DOI | MR | Zbl

[6] Birkhoff G.D., Langer R.E., “The boundary problems and developments associated with a system of ordinary differential equations of the first order”, Proc. Amer. Acad. Arts Sci., 58:2 (1923), 49–128 | DOI | Zbl

[7] Burlutskaya M.Sh., Kornev V.V., Khromov A.P., “Sistema Diraka s nedifferentsiruemym potentsialom i periodicheskimi kraevymi usloviyami”, ZhVMiMF, 52:9 (2012), 1621–1632 | Zbl

[8] Burlutskaya M.Sh., Kurdyumov V.P., Khromov A.P., “Utochnennye asimptoticheskie formuly dlya sobstvennykh znachenii i sobstvennykh funktsii sistemy Diraka”, DAN, 443:4 (2012), 414–417 | MR | Zbl

[9] Koddington E.A., Levinson N., Teoriya obyknovennykh differentsialnykh uravnenii, Izd-vo inostr. lit., M., 1958

[10] Dini U., Fondamenti per la teorica delle funzioni di variabili reali, Pisa, 1878

[11] Djakov P., Mityagin B., “Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions”, Indiana Univ. Math. J., 61:1 (2012), 359–398 | DOI | MR | Zbl

[12] Djakov P., Mityagin B., “Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators”, J. Funct. Anal., 263:8 (2012), 2300–2332 | DOI | MR | Zbl

[13] Djakov P., Mityagin B., “Riesz bases consisting of root functions of 1D Dirac operators”, Proc. Amer. Math. Soc., 141:4 (2013), 1361–1375 | DOI | MR | Zbl

[14] Djakov P., Mityagin B., “Riesz basis property of Hill operators with potentials in weighted spaces”, Tr. Mosk. mat. o-va., 75:2 (2014), 181–204 | MR | Zbl

[15] Gokhberg I.Ts., Krein M.G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR

[16] Gomilko A.M., Radzievskii G.V., “Ravnoskhodimost ryadov po sobstvennym funktsiyam obyknovennykh funktsionalno-differentsialnykh operatorov”, DAN SSSR, 316:2 (1991), 265–270 | MR | Zbl

[17] Haar A., “Zur Theorie der orthogonalen Funktionensysteme”, Math. Ann., 69:3 (1910), 331–371 ; 71:1 (1911), 38–53 | DOI | MR | Zbl | DOI | MR

[18] Khardi G.G., Littlvud Dzh.E., Polia G., Neravenstva, Izd-vo inostr. lit., M., 1948

[19] Ilin V.A., “Ravnoskhodimost s trigonometricheskim ryadom razlozhenii po kornevym funktsiyam odnomernogo operatora Shrëdingera s kompleksnym potentsialom iz klassa $L_1$”, Dif. uravneniya, 27:4 (1991), 577–597 | MR | Zbl

[20] Keldysh M.V., “O polnote sobstvennykh funktsii nekotorykh klassov nesamosopryazhennykh lineinykh operatorov”, UMN, 26:4 (1971), 15–41 | MR | Zbl

[21] Kornev V.V., Khromov A.P., “Sistema Diraka s nedifferentsiruemym potentsialom i antiperiodicheskimi kraevymi usloviyami”, Izv. Sarat. un-ta. Matematika. Mekhanika. Informatika, 13:3 (2013), 28–35 | Zbl

[22] Lomov I.S., “O lokalnoi skhodimosti biortogonalnykh ryadov, svyazannykh s differentsialnymi operatorami s negladkimi koeffitsientami. I”, Dif. uravneniya, 37:3 (2001), 328–342 | MR | Zbl

[23] Minkin A., “Equiconvergence theorems for differential operators”, J. Math. Sci., 96:6 (1999), 3631–3715 | DOI | MR | Zbl

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

[25] Nazarov A.I., Stolyarov D.M., Zatitskiy P.B., Tamarkin equiconvergence theorem and trace formula revisited, E-print, 2012, arXiv: ; Nazarov A.I., Stolyarov D.M., Zatitskiy P.B., “The Tamarkin equiconvergence theorem and a first-order trace formula for regular differential operators revisited.”, J. Spectr. Theory 4, 2014, no. 2, 365–389 1210.8097v2 [math.SP] | MR

[26] Sadovnichaya I.V., “O ravnoskhodimosti razlozhenii v ryady po sobstvennym funktsiyam operatorov Shturma–Liuvillya s potentsialami-raspredeleniyami”, Mat. sb., 201:9 (2010), 61–76 | DOI | MR | Zbl

[27] Sadovnichaya I.V., “Equiconvergence theorems for Sturm–Liouville operators with singular potentials (rate of equiconvergence in $W_2^\theta $-norm)”, Eurasian Math. J., 1:1 (2010), 137–146 | MR | Zbl

[28] Sadovnichaya I.V., “Ravnoskhodimost v prostranstvakh Gëldera razlozhenii po sobstvennym funktsiyam operatorov Shturma–Liuvillya s potentsialami-raspredeleniyami”, Dif. uravneniya, 48:5 (2012), 674–685 | MR | Zbl

[29] Savchuk A.M., Sadovnichaya I.V., “Bazisnost Rissa so skobkami dlya sistemy Diraka s summiruemym potentsialom”, Sovr. matematika. Fund. napr., 58 (2015), 128–152

[30] Savchuk A.M., Shkalikov A.A., “The Dirac operator with complex-valued summable potential”, Math. Notes, 96:5 (2014), 777–810 | DOI | MR | Zbl

[31] Scherbakov A.O., “Regulyarizovannyi sled operatora Diraka”, Mat. zametki, 98:1 (2015), 134–146 | DOI | MR

[32] Shveikina O.A., “Teoremy o ravnoskhodimosti dlya singulyarnykh operatorov Shturma–Liuvillya s razlichnymi kraevymi usloviyami”, Dif. uravneniya, 51:2 (2015), 174–182 | DOI | MR | Zbl

[33] Stekloff W., “Sur les expressions asymptotiques de certaines fonctions, définies par les équations différentielles linéaires du second ordre, et leurs applications au problème du développement d'une fonction arbitraire en séries procédant suivant les dites fonctions”, Soobsch. Kharkov. mat. o-va. Ser. 2, 10 (1907), 97–201; Стеклов В.А., Об асимптотическом выражении некоторых функций, определяемых линейным дифференциальным уравнением второго порядка, и их применении к задаче разложения произвольной функции в ряд по этим функциям, Изд-во Харьков. ун-та, Харьков, 1956

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

[35] Tamarkin Ya.D., O nekotorykh obschikh zadachakh teorii obyknovennykh lineinykh differentsialnykh uravnenii i o razlozhenii proizvolnykh funktsii v ryady, Tip. M.P. Frolovoi, Petrograd, 1917

[36] Tamarkin J., “Some general problems of the theory of linear differential equations and expansion of an arbitrary function in series of fundamental functions”, Math. Z., 27:1 (1928), 1–54 | DOI | MR

[37] Vinokurov V.A., Sadovnichii V.A., “Ravnomernaya ravnoskhodimost ryada Fure po sobstvennym funktsiyam pervoi kraevoi zadachi i trigonometricheskogo ryada Fure”, DAN, 380:6 (2001), 731–735 | MR | Zbl