Voir la notice de l'article provenant de la source Math-Net.Ru
@article{CMFD_2020_66_3_a0, author = {A. M. Savchuk and I. V. Sadovnichaya}, title = {Spectral analysis of one-dimensional {Dirac} system with summable potential and {Sturm--Liouville} operator with distribution coefficients}, journal = {Contemporary Mathematics. Fundamental Directions}, pages = {373--530}, publisher = {mathdoc}, volume = {66}, number = {3}, year = {2020}, language = {ru}, url = {https://geodesic-test.mathdoc.fr/item/CMFD_2020_66_3_a0/} }
TY - JOUR AU - A. M. Savchuk AU - I. V. Sadovnichaya TI - Spectral analysis of one-dimensional Dirac system with summable potential and Sturm--Liouville operator with distribution coefficients JO - Contemporary Mathematics. Fundamental Directions PY - 2020 SP - 373 EP - 530 VL - 66 IS - 3 PB - mathdoc UR - https://geodesic-test.mathdoc.fr/item/CMFD_2020_66_3_a0/ LA - ru ID - CMFD_2020_66_3_a0 ER -
%0 Journal Article %A A. M. Savchuk %A I. V. Sadovnichaya %T Spectral analysis of one-dimensional Dirac system with summable potential and Sturm--Liouville operator with distribution coefficients %J Contemporary Mathematics. Fundamental Directions %D 2020 %P 373-530 %V 66 %N 3 %I mathdoc %U https://geodesic-test.mathdoc.fr/item/CMFD_2020_66_3_a0/ %G ru %F CMFD_2020_66_3_a0
A. M. Savchuk; I. V. Sadovnichaya. Spectral analysis of one-dimensional Dirac system with summable potential and Sturm--Liouville operator with distribution coefficients. Contemporary Mathematics. Fundamental Directions, Spectral Analysis, Tome 66 (2020) no. 3, pp. 373-530. https://geodesic-test.mathdoc.fr/item/CMFD_2020_66_3_a0/
[1] F. Atkinson, Discrete and Continuous Boundary-Value Problems, Russian translation, Mir, M., 1968
[2] A. G. Baskakov, T. K. Katsaryan, “Spectral analysis of integrodifferential operators with nonlocal boundary conditions”, Differ. Equ., 24:8 (1988), 1424–1433 (in Russian) | MR | Zbl
[3] A. A. Belyaev, A. A. Shkalikov, “Multipliers in spaces of Bessel potentials: the case of indices of nonnegative smoothness”, Math. Notes, 102:5 (2017), 684–699 (in Russian) | MR | Zbl
[4] J. Bergh, J. Lefstrem, Interpolation Spaces, Russian translation, Mir, M., 1980
[5] V. I. Bogachev, O. G. Smolyanov, Real and Functional Analysis, NITs Regulyarnaya i khaoticheskaya dinamika, M.–Izhevsk, 2009 (in Russian)
[6] M. Sh. Burlutskaya, V. V. Kornev, A. P. Khromov, “Dirac system with nondifferentiable potential and periodic boundary conditions”, J. Comput. Math. Math. Phys., 52:9 (2012), 1621–1632 (in Russian) | Zbl
[7] M. Sh. Burlutskaya, V. P. Kurdyumov, A. P. Khromov, “Refined asymptotic formulas for eigenvalues and eigenfunctions of the Dirac system”, Rep. Russ. Acad. Sci., 443:4 (2012), 414–417 (in Russian) | MR | Zbl
[8] A. I. Vagabov, “On refinement of the Tamarkin asymptotic theorem”, Differ. Equ., 29:1 (1993), 41–49 (in Russian) | MR | Zbl
[9] A. I. Vagabov, “Ob asimptotike po parametru resheniy differentsial'nykh sistem s koeffitsientami iz klassa $L_q$”, Differ. Equ., 46:1 (2010), 16–22 (in Russian) | MR | Zbl
[10] O. A. Veliev, A. A. Shkalikov, “On the Riesz basis property of eigen and associated functions of periodic and antiperiodic Sturm-Liouville problems”, Math. Notes, 85:5 (2009), 671–686 (in Russian) | MR | Zbl
[11] V. A. Vinokurov, V. A. Sadovnichiy, “Asymptotics of eigenvalues and eigenfunctions and the trace formula for a potential containing $\delta$-functions”, Differ. Equ., 38:6 (2002), 735–751 (in Russian) | MR | Zbl
[12] A. A. Vladimirov, “On convergence of sequences of ordinary differential operators”, Math. Notes, 75:6 (2004), 941–943 (in Russian) | Zbl
[13] A. A. Vladimirov, I. A. Sheypak, Proc. Math. Inst. Russ. Acad. Sci., 255 (2006), 88–98 (in Russian) | Zbl
[14] A. A. Vladimirov, I. A. Sheypak, “Self-similar functions in the space $L_2[0, 1]$ and the Sturm–Liouville problem with a singular indefinite weight”, Math. Digest, 197:11 (2006), 13–30 (in Russian) | MR | Zbl
[15] A. A. Vladimirov, I. A. Sheypak, “On the Neumann problem for the Sturm-Liouville equation with self-similar weight of Cantor type”, Funct. Anal. Appl., 47:4 (2013), 18–29 (in Russian) | MR | Zbl
[16] V. E. Vladykina, “Spectral characteristics of the Sturm-Liouville operator under minimal conditions on the smoothness of the coefficients”, Bull. Moscow Univ. Ser. 1. Math. Mech., 2019, no. 6, 23–28 (in Russian) | MR | Zbl
[17] J. Garnett, Bounded Analytic Functions, Russian translation, Mir, M., 1984
[18] A. M. Gomilko, G. V. Radzievskiy, “Equiconvergence of series in eigenfunctions of ordinary functional differential operators”, Rep. Russ. Acad. Sci., 316:2 (1991), 265–270 (in Russian) | Zbl
[19] I. Gokhberg, M. G. Kreyn, Introduction to the Theory of Linear Non-self-adjoint Operators in Hilbert Space, Nauka, M., 1965 (in Russian)
[20] N. Danford, D. Shvarts, Lineinye operatory, v. I, Mir, M., 1962; т. II, 1966; т. III, 1974 [N. Dunford, J. Schwartz, Linear Operators, v. I, Mir, M., 1962]; v. II, 1966; Russian translation, v. III, 1974
[21] V. A. Il'in, “Necessary and sufficient conditions for spectral expansions to have the basis property and equiconvergence with trigonometric series. I”, Differ. Equ., 16:5 (1980), 771–794 (in Russian) | MR | Zbl
[22] V. A. Il'in, “Necessary and sufficient conditions for spectral expansions to have the basis property and equiconvergence with trigonometric series. II”, Differ. Equ., 16:6 (1980), 980–1009 (in Russian) | MR | Zbl
[23] V. A. Il'in, “Equiconvergence with trigonometric series of expansions in root functions of the one-dimensional Schrodinger operator with complex potential of class $L_1$”, Differ. Equ., 27:4 (1991), 577–597 (in Russian) | MR | Zbl
[24] V. A. Il'in, “Uniform on the whole axis $\mathbb{R}$ equiconvergence with the Fourier integral of the spectral expansion corresponding to self-adjoint extension of the Schrodinger operator with uniformly locally summable potential”, Differ. Equ., 31:12 (1995), 1957–1967 (in Russian) | MR | Zbl
[25] V. A. Il'in, I. Antoniu, “On the uniform on the whole axis $\mathbb{R}$ equiconvergence with the Fourier integral of the spectral expansion of an arbitrary function from $L_p(\mathbb{R})$ corresponding to self-adjoint extension of the Hill operator”, Differ. Equ., 31:8 (1995), 1310–1322 (in Russian) | MR | Zbl
[26] V. A. Il'in, I. Antoniu, “On spectral expansions corresponding to the Liouville operator generated by the Schrodinger operator with uniformly locally summable potential”, Differ. Equ., 32:4 (1996), 435–440 (in Russian) | MR | Zbl
[27] T. Kato, Perturbation Theory for Linear Operators, Russian translation, Mir, M., 1972
[28] I. S. Kats, “On the existence of spectral functions of some second-order singular differential systems”, Rep. Acad. Sci. USSR, 106:1 (1956), 15–18 (in Russian) | Zbl
[29] I. S. Kats, M. G. Kreyn, Addition II “On spectral functions of a string” to the book by F. Atkinson “Discrete and Continuous Boundary-Value Problems”, Mir, M., 1968 (in Russian)
[30] V. E. Katsnel-son, “On basicity conditions of a system of root vectors of some classes of operators”, Funct. Anal. Appl., 1:2 (1967), 39–51 (in Russian) | MR
[31] M. V. Keldysh, “On completeness of eigenfunctions of some classes of non-self-adjoint linear operators”, Progr. Math. Sci., 26:4 (1971), 15–41 (in Russian) | MR | Zbl
[32] G. M. Kesel'man, “On unconditional convergence of expansions in eigenfunctions of specific differential operators”, Bull. Higher Edu. Inst. Ser. Math., 39:2 (1964), 82–93 (in Russian) | Zbl
[33] E. Coddington, N. Levinson, Theory of Ordinary Differential Equations, Russian translation, Izd-vo inostrannoy literatury, M., 1958
[34] V. V. Kornev, A. P. Khromov, “Dirac system with nondifferentiable potential and antiperiodic boundary conditions”, Bull. Saratov Univ. N.S. Ser. Math. Mech. Inform., 13:3 (2013), 28–35 (in Russian) | Zbl
[35] A. G. Kostyuchenko, A. A. Shkalikov, “On the summability of the eigenfunction expansions of differential operators and convolution operators”, Funct. Anal. Appl., 12:4 (1978), 24–40 (in Russian) | MR | Zbl
[36] M. A. Krasnoselskii, P. P. Zabreyko, E. I. Pustyl'nik, P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Nauka, M., 1966 (in Russian) | MR
[37] M. G. Kreyn, “On the indefinite case of the Sturm-Liouville boundary-value problem in the interval $(0,\infty)$”, Bull. Russ. Acad. Sci. Ser. Math., 16:4 (1952), 293–324 (in Russian) | Zbl
[38] B. M. Levitan, “On the asymptotic behavior of the spectral function of a second-order self-adjoint differential equation and on the expansion in eigenfunctions”, Bull. Acad. Sci. USSR Ser. Math., 17 (1953), 331–367 (in Russian)
[39] B. M. Levitan, “On the asymptotic behavior of the spectral function of a second-order self-adjoint differential equation and on the expansion in eigenfunctions. II”, Bull. Acad. Sci. USSR Ser. Math., 19:1 (1955), 33–58 (in Russian) | Zbl
[40] B. M. Levitan, I. S. Sargsyan, Sturm-Liouville and Dirac Operators, Nauka, M., 1988 (in Russian) | MR
[41] B. V. Lidskiy, “On summability of series in principal vectors of non-self-adjoint operators”, Proc. Moscow Math. Soc., 11, 1962, 3–35 (in Russian)
[42] I. S. Lomov, “On local convergence of biorthogonal series associated with differential operators with nonsmooth coefficients. I, II”, Differ. Equ., 37:3 (2001), 328–342 (in Russian) | MR | Zbl
[43] A. A. Lunev, M. M. Malamud, “On the Riesz basis property of a system of root vectors for a $2\times 2$-system of Dirac type”, Rep. Russ. Acad. Sci., 458:3 (2014), 1–6 (in Russian)
[44] A. S. Makin, “On convergence of expansions in root functions of a periodic boundary-value problem”, Rep. Russ. Acad. Sci., 406:4 (2006), 452–457 (in Russian) | MR | Zbl
[45] A. S. Markus, “On the expansion in root vectors of a weakly perturbed self-adjoint operator”, Rep. Acad. Sci. USSR, 142:3 (1962), 538–541 (in Russian) | Zbl
[46] V. A. Marchenko, “Some questions of the theory of one-dimensional second-order linear differential operators. I”, Proc. Moscow Math. Soc., 1, 1952, 327–420 (in Russian)
[47] V. A. Marchenko, “Some questions of the theory of one-dimensional second-order linear differential operators. II”, Proc. Moscow Math. Soc., 1, 1953, 3–83 (in Russian)
[48] K. A. Mirzoev, “Sturm-Liouville operators”, Proc. Moscow Math. Soc., 75:2 (2014), 335–359 (in Russian) | Zbl
[49] K. A. Mirzoev, A. A. Shkalikov, “Differential operators of even order with distribution coefficients”, Math. Notes, 99:5 (2016), 788–793 (in Russian) | MR | Zbl
[50] V. P. Mikhaylov, “On the Riesz basis property in $L_2(0, 1)$”, Rep. Acad. Sci. USSR, 144 (1962), 981–984 (in Russian)
[51] M. A. Naymark, Linear Differential Operators, Nauka, M., 1969 (in Russian)
[52] M. I. Neiman-Zade, A. A. Shkalikov, “Schrodinger operators with singular potentials from spaces of multipliers”, Math. Notes, 66:5 (1999), 723–733 (in Russian) | MR | Zbl
[53] I. M. Rapoport, On Some Asymptotic Methods in the Theory of Differential Equations, Izd-vo Akad. Nauk Ukr. SSR, Kiev, 1954 (in Russian) | MR
[54] F. Riesz, B. Szokefalvi-Nagy, Functional Analyzis, Russian translation, Mir, M., 1979
[55] V. S. Rykhlov, “Asymptotics of a system of solutions for a quasi-differential operator”, Differ. Equations and Theory of Functions, 5, Saratov Univ., Saratov, 1983, 51–59 (in Russian)
[56] V. S. Rykhlov, “On the rate of equiconvergence for differential operators with nonzero coefficient at the $(n-1)$th derivative”, Rep. Acad. Sci. USSR, 279:5 (1984), 1053–1056 (in Russian) | MR | Zbl
[57] A. M. Savchuk, “The Dirac system with potential from Besov spaces”, Differ. Equ., 52:4 (2016), 454–469 (in Russian) | Zbl
[58] A. M. Savchuk, “Calderon-Zygmund type operator and its connection with asymptotic estimates for ordinary differential operators”, Contemp. Math. Fundam. Directions, 63, no. 4, 2017, 689–702 (in Russian) | MR
[59] A. M. Savchuk, “On the basis property of the system of eigenfunctions and associated functions of the one-dimensional Dirac operator”, Bull. Russ. Acad. Sci. Ser. Math., 82:2 (2018), 113–139 (in Russian) | MR | Zbl
[60] A. M. Savchuk, “Uniform residual estimates arising in spectral analysis of linear differential systems”, Differ. Equ., 55:5 (2019), 625–635 (in Russian) | Zbl
[61] A. M. Savchuk, I. V. Sadovnichaya, “Asymptotic formulas for fundamental solutions of the Dirac system with a complex-valued summable potential”, Differ. Equ., 49:5 (2013), 573–584 (in Russian) | Zbl
[62] A. M. Savchuk, I. V. Sadovnichaya, “Riesz basis property from subspaces for the Dirac system with summable potential”, Rep. Russ. Acad. Sci., 462:3 (2015), 274–277 (in Russian) | Zbl
[63] A. M. Savchuk, I. V. Sadovnichaya, “Riesz basis property with brackets for the Dirac system with summable potential”, Contemp. Math. Fundam. Directions, 58, 2015, 128–152 (in Russian)
[64] A. M. Savchuk, I. V. Sadovnichaya, “Uniform basis property of the system of root vectors of the Dirac operator”, Contemp. Math. Fundam. Directions, 64, no. 1, 2018, 180–193 (in Russian) | MR
[65] A. M. Savchuk, A. A. Shkalikov, “Sturm-Liouville operators with singular potentials”, Math. Notes, 66:6 (1999), 897–912 (in Russian) | Zbl
[66] A. M. Savchuk, A. A. Shkalikov, “The trace formula for Sturm-Liouville operators with singular potentials”, Math. Notes, 68:3 (2000), 427–442 (in Russian)
[67] A. M. Savchuk, A. A. Shkalikov, “Sturm-Liouville operators with distribution potentials”, Proc. Mosk. mat. Moscow Math. Soc., 64 (2003), 159–219 (in Russian)
[68] A. M. Savchuk, A. A. Shkalikov, “On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces”, Math. Notes, 80:6 (2006), 864–884 (in Russian) | Zbl
[69] A. M. Savchuk, A. A. Shkalikov, “Inverse problems for the Sturm-Liouville operator with potentials from Sobolev spaces. Uniform stability”, Funct. Anal. Appl., 44:4 (2010), 34–53 (in Russian) | MR | Zbl
[70] A. M. Savchuk, A. A. Shkalikov, “Uniform stability of the inverse Sturm-Liouville problem with respect to the spectral function in the scale of Sobolev spaces”, Proc. Math. Inst. Russ. Acad. Sci., 283 (2013), 188–203 (in Russian) | Zbl
[71] A. M. Savchuk, A. A. Shkalikov, “Asymptotic analysis of solutions of ordinary differential equations with distribution coefficients”, Math. Digest, 2020 (in Russian)
[72] I. V. Sadovnichaya, “On equiconvergence of expansions in series of eigenfunctions of Sturm-Liouville operators with distributions potentials”, Math. Digest, 201:9 (2010), 61–76 (in Russian) | MR | Zbl
[73] I. V. Sadovnichaya, “Equiconvergence of spectral expansions for the Dirac system with potential from Lebesgue spaces”, Proc. Math. Inst. Russ. Acad. Sci., 293 (2016), 296–324 (in Russian) | Zbl
[74] I. V. Sadovnichaya, “Convergence of spectral expansions for the Sturm–Liouville operator”, Abstracts of Int. Conf. on Differ. Equ. and Dynam. Systems (Suzdal', Russia, 6–11 July 2018), Izd-vo VlGU, Vladimir, 2018, 185–186 (in Russian)
[75] I. V. Sadovnichaya, “Equiconvergence of spectral expansions for second-order ordinary differential operators with distribution coefficients”, Abstracts of Int. Conf. on Differ. Equ. and Dynam. Systems (Suzdal', Russia, 3–8 July 2020), Izd-vo VlGU, Vladimir, 2020, 107–108 (in Russian)
[76] A. M. Sedletskiy, “O ravnomernoy skhodimosti negarmonicheskikh ryadov Fur'e”, Proc. Math. Inst. Russ. Acad. Sci., 200, 1991, 299–309 (in Russian)
[77] V. A. Steklov, “Sur les expressions asymptotiques de certaines fonctions, définies par les équations differentielles linéaires du second ordre, et leurs applications au problème du développement d'une fonction arbitraire en séries procédant suivant les-dites fonctions”, Soobshch. Khar'kov. mat. ob-va. Vtoraya ser., 10 (1907), 97–199 (in Russian)
[78] Ya. D. Tamarkin, On Some General Problems in the Theory of Ordinary Differential Equations and on the Expansion of an Arbitrary Function in Series, Tipografiya Frolovoy, Petrograd, 1917 (in Russian)
[79] H. Tribel, Interpolation Theory, Function Spaces, Differential Operators, Russian translation, Mir, M., 1980
[80] H. Tribel, Theory of Function Spaces, Mir, M., 1986 (in Russian)
[81] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Russian translation, Gos. izd-vo inostrannoy literatury, M., 1948
[82] A. P. Khromov, “Expansion in eigenfunctions of ordinary differential operators on a finite interval”, Rep. Acad. Sci. USSR, 146:6 (1962), 1294–1297 (in Russian)
[83] A. P. Khromov, “Expansion in eigenfunctions of ordinary linear differential operators with irregular separating boundary conditions”, Math. Digest, 70 (1966), 310–329 (in Russian)
[84] A. P. Khromov, “On summation of expansions in eigenfunctions of a boundary-value problem for an ordinary differential equation with separating boundary conditions and an analogue of the Weierstrass theorem”, Ordinary Differential Equations and Fourier Series Expansions, Saratovskiy un-t, Saratov, 1968, 29–41 (in Russian)
[85] A. P. Khromov, “On equiconvergence of expansions in eigenfunctions of second-order differential operators. II”, Differential Equations and Computational Mathematics, 5, Izd-vo Saratovskogo un-ta, Saratov, 1975, 3–20 (in Russian)
[86] D. Shin, “Existence theorem for an $n$th-order quasi-differential equation”, Rep. Acad. Sci. USSR, 18:8 (1938), 515–518 (in Russian)
[87] D. Shin, “On solutions of linear $n$-th-order quasi-differential equation”, Math. Digest, 7:3 (1940), 479–532 (in Russian)
[88] D. Shin, “On quasi-differential operators in a Hilbert space”, Math. Digest, 13:1 (1943), 39–70 (in Russian) | Zbl
[89] A. A. Shkalikov, “On the basis property of eigenfunctions of an ordinary differential operator”, Progr. Math. Sci., 34:5 (1979), 235–236 (in Russian) | MR | Zbl
[90] A. A. Shkalikov, “On the basis property of eigenvectors of quadratic operator pencils”, Math. Notes, 30:3 (1981), 371–385 (in Russian) | MR | Zbl
[91] A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Progr. Math. Sci., 71:5 (2016), 113–174 (in Russian) | MR | Zbl
[92] V. A. Yurko, Introduction to the Theory of Inverse Spectral Problems, Fizmatlit, M., 2007 (in Russian)
[93] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable models in quantum mechanics, AMS Chelsea Publishing, Providence, 2005 | MR | Zbl
[94] S. Albeverio, P. Kurasov, Singular perturbations of differential operators, Cambridge Univ. Press, Cambridge, 2001 | MR
[95] F. Atkinson, W. Everitt, A. Zettl, “Regularization of a Sturm-Liouville problem with an interior singularity using quasi-derivatives”, Differ. Integral Equ., 1:2 (1988), 213–221 | MR | Zbl
[96] J. G. Bak, A. A. Shkalikov, “Multipliers in dual Sobolev spaces and Schrodinger operators with distribution potentials”, Math. Notes, 71 (2002), 587–594 | DOI | MR | Zbl
[97] A. G. Baskakov, D. M. Polyakov, “Spectral properties of the Hill operator”, Math. Notes, 99:3-4 (2016), 598–602 | DOI | MR | Zbl
[98] A. G. Baskakov, D. M. Polyakov, “The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential”, Sb. Math., 208:1 (2017), 1–43 | DOI | MR | Zbl
[99] J. Ben Amara, A. A. Shkalikov, “Oscillation theorems for Sturm-Liouville problems with distribution potentials”, Moscow Univ. Math. Bull., 64:3 (2009), 132–137 | DOI | MR | Zbl
[100] C. Bennett, R. C. Sharpley, Interpolation of operators, Academic press, Boston etc., 1988 | MR | Zbl
[101] C. Bennewitz, “Spectral asymptotics for Sturm-Liouville equations”, Proc. Lond. Math. Soc. (3), 59:2 (1989), 294–338 | DOI | MR | Zbl
[102] C. Bennewitz, W. N. Everitt, “On second-order left-definite boundary value problems”, Ordinary differential equations and operators. A tribute to F. V. Atkinson, Symp. (Dundee, Scotland, March–July 1982), Springer-Verlag, Berlin etc., 1983, 31–67 | DOI | MR
[103] H. E. Benzinger, “Green's function for ordinary differential operators”, J. Differ. Equ., 7:3 (1970), 478–496 | DOI | MR | Zbl
[104] G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter”, Trans. Am. Math. Soc., 9:2 (1908), 219–231 | DOI | MR | Zbl
[105] G. D. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations”, Trans. Am. Math. Soc., 9:4 (1908), 373–395 | DOI | MR | Zbl
[106] R. Camassa, D. Holm, “An integrable shallow water equation with peaked solitons”, Phys. Rev. Lett., 71 (1993), 1661–1664 | DOI | MR | Zbl
[107] P. Djakov, B. Mityagin, “Instability zones of periodic 1-dimensional Schrodinger and Dirac operators”, Russ. Math. Surv., 61:4 (2006), 663–766 | DOI | MR | Zbl
[108] P. Djakov, B. Mityagin, “Bari-Markus property for Riesz projections of Hill operators with singular potentials”, Contemp. Math., 481, 2009, 59–80 | DOI | MR | Zbl
[109] P. Djakov, B. Mityagin, “Spectral gap asymptotics of one-dimensional Schrodinger operators with singular periodic potentials”, Integral Transforms Spec. Funct., 20:3-4 (2009), 265–273 | DOI | MR | Zbl
[110] P. Djakov, B. Mityagin, “Fourier method for one-dimensional Schrodinger operators with singular periodic potentials”, Topics in operator theory, IWOTA, Proc. 19th Int. Workshop Operator Theory Appl. (Williamsburg, USA, July 22–26, 2008), v. 2, Systems and mathematical physics, Birkhauser, Basel, 2010, 195–236 | DOI | MR
[111] P. Djakov, B. Mityagin, “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
[112] P. Djakov, B. Mityagin, “Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions”, J. Approx. Theory, 164:7 (2012), 879–927 | DOI | MR | Zbl
[113] P. Djakov, B. Mityagin, “Equiconvergence of spectral decompositions of Hill operators”, Dokl. Math., 86:1 (2012), 542–544 | DOI | MR | Zbl
[114] P. Djakov, B. Mityagin, “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
[115] P. Djakov, B. Mityagin, “Equiconvergence of spectral decompositions of Hill-Schrodinger operators”, J. Differ. Equ., 255:10 (2013), 3233–3283 | DOI | MR | Zbl
[116] P. Djakov, B. Mityagin, “Riesz basis property of Hill operators with potentials in weighted spaces”, Trans. Moscow Math. Soc., 75 (2014), 151–172 | DOI | MR | Zbl
[117] N. Dunford, “A survey of the thoery of spectral operators”, Bull. Am. Math. Soc. (N.S.), 64 (1958), 217–274 | DOI | MR | Zbl
[118] J. Eckhardt, A. Kostenko, “The inverse spectral problem for indefinite strings”, Invent. Math., 204:3 (2016), 939–977 | DOI | MR | Zbl
[119] J. Eckhardt, A. S. Kostenko, M. M. Malamud, G. Teschl, “One-dimensional Schrodinger operators with $\delta'$-interactions on Cantor-type sets”, J. Differ. Equ., 257 (2014), 415–449 | DOI | MR | Zbl
[120] J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl, “Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials”, Opuscula Math., 33:3 (2013), 467–563 | DOI | MR | Zbl
[121] J. Eckhardt, G. Teschl, “Sturm-Liouville operators with measure-valued coefficients”, J. d'Anal. Math., 120:1 (2013), 151–224 | DOI | MR | Zbl
[122] W. N. Everitt, L. Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, Amer. Math. Soc., Providence, 1999 | MR | Zbl
[123] W. Feller, “Generalized second order differential operators and their lateral conditions”, Illinois J. Math., 1:4 (1957), 459–504 | DOI | MR | Zbl
[124] C. Frayer, R. O. Hryniv, Ya. V. Mykytyuk, P. A. Perry, “Inverse scattering for Schrodinger operators with Miura potentials. I. Unique Riccati representatives and ZS-AKNS system”, Inverse Problems, 25:11 (2009), 115007 | DOI | MR | Zbl
[125] L. Grafakos, Modern Fourier analysis, Springer, New York, 2009 | MR | Zbl
[126] S. Grudsky, A. Rybkin, “On positive type initial profiles for the KdV equation”, Proc. Am. Math. Soc., 142:6 (2014), 2079–2086 | DOI | MR | Zbl
[127] J. Gunson, “Perturbation theory for a Sturm-Liouville problem with an interior singularity”, Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci., 414 (1987), 255–269 | MR | Zbl
[128] A. Haar, “Zur Theorie der orthogonalen Funktionensysteme. (Erste Mitteilung.)”, Math. Ann., 69:3 (1910), 331–371 | DOI | MR | Zbl
[129] A. Haar, “Zur Theorie der orthogonalen Funktionensysteme. (Zweite Mitteilung.)”, Math. Ann., 71:1 (1912), 38–53 | DOI | MR | Zbl
[130] E. W. Hobson, “On a general convergence theorem, and the theory of the representation of a function by a series of normal functions”, Proc. Lond. Math. Soc. (2), 6 (1908), 349–395 | DOI | MR | Zbl
[131] R. Hryniv, Ya. Mykytyuk, “1D Schrodinger operators with singular periodic potentials”, Methods Funct. Anal. Topol., 7:4 (2001), 31–42 | MR | Zbl
[132] R. Hryniv, Ya. Mykytyuk, “Inverse spectral problems for Sturm-Liouville operators with singular potentials”, Inverse Problems, 19:3 (2003), 665–684 | DOI | MR | Zbl
[133] R. Hryniv, Ya. Mykytyuk, “Inverse spectral problems for Sturm-Liouville operators with singular potentials. III. Reconstruction by three spectra”, J. Math. Anal. Appl., 284:2 (2003), 626–646 | DOI | MR | Zbl
[134] R. Hryniv, Ya. Mykytyuk, “Transformation operators for Sturm-Liouville operators with singular potentials”, Math. Phys. Anal. Geom., 7:2 (2004), 119–149 | DOI | MR | Zbl
[135] R. Hryniv, Ya. Mykytyuk, “Inverse spectral problems for Sturm-Liouville operators with singular potentials. IV. Potentials in the Sobolev space scale”, Proc. Edinb. Math. Soc. (2), 49:2 (2006), 309–329 | DOI | MR | Zbl
[136] R. Hryniv, Ya. Mykytyuk, P. A. Perry, “Inverse scattering for Schrodinger operators with Miura potentials. II. Different Riccati representatives”, Commun. Part. Differ. Equ., 36 (2011), 1587–1623 | DOI | MR | Zbl
[137] R. Hryniv, Ya. Mykytyuk, P. A. Perry, “Sobolev mapping properties of the scattering transform for the Schrodinger equation”, Spectral theory and geometric analysis, Int. Conf. in honor of Mikhail Shubin's 65th birthday (Boston, USA, July 29–August 2, 2009), Am. Math. Soc., Providence, 2011, 79–93 | DOI | MR | Zbl
[138] T. Kappeler, C. Mohr, “Estimates for periodic and Dirichlet eigenvalues of the Schrodinger operator with singular potentials”, J. Funct. Anal., 186 (2001), 62–91 | DOI | MR | Zbl
[139] T. Kappeler, P. Topalov, “Global well-posedness of mKdV in $L_2(\mathbb{T}, \mathbb{R})$”, Commun. Part. Differ. Equ, 30 (2005), 435–449 | DOI | MR | Zbl
[140] T. Kappeler, P. Topalov, “Global wellposedness of KdV in $H^{-1}(\mathbb{R}, \mathbb{R})$”, Duke Math. J., 135 (2006), 327–360 | DOI | MR | Zbl
[141] B. S. Kashin, A. A. Saakyan, Orthogonal series, Am. Math. Soc., Providence, 1989 | MR | Zbl
[142] E. Korotyaev, “Characterization of the spectrum of Schrodinger operators with periodic distributions”, Int. Math. Res. Not. IMRN, 37 (2003), 2019–2031 | DOI | MR | Zbl
[143] A. S. Kostenko, M. M. Malamud, “1-D Schrodinger operators with local point interactions on a discrete set”, J. Differ. Equ, 249 (2010), 253–304 | DOI | MR | Zbl
[144] A. S. Kostenko, M. M. Malamud, “One-dimensional Schrodinger operator with $\delta$-interactions”, Funct. Anal. Appl., 44:2 (2010), 151–155 | DOI | MR | Zbl
[145] P. Kurasov, “On the Coulomb potentials in one dimension”, J. Phys. A, 29:8 (1996), 1767–1771 | DOI | MR | Zbl
[146] H. Langer, Zur Spektraltheorie verallgemeinerter gewonlicher Differentialoperatoren zweiter Ordnung mit einer nichtmonotonen Gewichtsfunktion, Univ. Jyväskylä Math. Inst., Jyvaskylä, 1972 | MR
[147] A. A. Lunyov, M. M. Malamud, “On the completeness of the root vectors for first order systems”, Dokl. Math., 88:3 (2013), 678–683 | DOI | MR | Zbl
[148] A. A. Lunyov, M. M. Malamud, “On the Riesz basis property of root vectors system for $2\times 2$ Dirac type operators”, J. Math. Anal. Appl., 441 (2016), 57–103 | DOI | MR | Zbl
[149] M. M. Malamud, L. L. Oridoroga, “On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations”, J. Funct. Anal., 263 (2012), 1939–1980 | DOI | MR | Zbl
[150] V. G. Maz'ya, I. E. Verbitsky, “Boundedness and compactness criteria for the one-dimensional Schrodinger operator”, Function spaces, interpolation theory and related topics, Proc. Int. Conf. in honour of J. Peetre on his 65th birthday (Lund, Sweden, August 17-22, 2000), de Gruyter, Berlin, 2002, 369–382 | MR | Zbl
[151] V. G. Maz'ya, I. E. Verbitsky, “The form boundedness criterion for the relativistic Schrodinger operator”, Ann. Inst. Fourier (Grenoble), 54:2 (2004), 317–339 | DOI | MR | Zbl
[152] V. G. Maz'ya, I. E. Verbitsky, “Infinitesimal form boundedness and Trudinger's subordination for the Schrodinger operator”, Invent. Math., 162 (2005), 81–136 | DOI | MR | Zbl
[153] V. A. Mikhailets, V. M. Molyboga, “Singular eigenvalue problems on the circle”, Methods Funct. Anal. Topol., 10:3 (2004), 44–53 | MR | Zbl
[154] V. A. Mikhailets, V. M. Molyboga, “One-dimensional Schrodinger operators with singular periodic potentials”, Methods Funct. Anal. Topol., 14:2 (2008), 184–200 | MR | Zbl
[155] V. A. Mikhailets, V. M. Molyboga, “Spectral gaps of the one-dimensional Schrodinger operators with singular periodic potentials”, Methods Funct. Anal. Topol., 15:1 (2009), 31–40 | MR | Zbl
[156] A. B. Mingarelli, Volterra-Stieltjes integral equations and generalized ordinary differential expressions, Springer, Berlin, 1983 | MR | Zbl
[157] A. Minkin, “Equiconvergence theorems for differential operators”, J. Math. Sci. (N. Y.), 96 (1999), 3631–3715 | DOI | MR | Zbl
[158] G. V. Radzievskii, “Boundary value problems and related moduli of continuity”, Funct. Anal. Appl., 29:3 (1995), 217–219 | DOI | MR
[159] A. Rybkin, “Regularized perturbation determinants and KdV conservation laws for irregular initial profiles”, Topics in operator theory, IWOTA, Proc. 19th Int. Workshop Operator Theory Appl. (Williamsburg, USA, July 22–26, 2008), v. 2, Systems and mathematical physics, Birkhauser, Basel, 2010, 427–444 | DOI | MR | Zbl
[160] V. S. Rykhlov, “Asymptotical formulas for solutions of linear differential systems of the first order”, Results Math., 36 (1999), 342–353 | DOI | MR | Zbl
[161] A. M. Savchuk, A. A. Shkalikov, “The Dirac operator with complex-valued summable potential”, Math. Notes, 96:5 (2014), 3–36 | MR
[162] V. A. Stekloff, “Solution genérale du probl eme de dèveloppement d'une fonction arbitraire en séries suivant les fonctions fondamentales de Sturm-Liouville”, Rom. Acc. L. Rend. (5), 19 (1910), 490–496 | Zbl
[163] M. H. Stone, “A comparison of the series of Fourier and Birkhoff”, Trans. Am. Math. Soc., 28:4 (1926), 695–761 | DOI | MR
[164] J. D. Tamarkine, “Application de la methode des fonctions fondamentales a l'ètude de l'équation différentielle des verges vibrantes élastiques”, Rep. Kharkov Math. Soc. 2nd Ser., 12 (1911), 19–46
[165] J. D. Tamarkine, “Addition à l'article intitulé «Application de la méthode des fonctions fondamentales à l'étude de l'équation différentielle des verges vibrantes élastiques»”, Rep. Kharkov Math. Soc. 2nd Ser., 12 (1911), 65–69
[166] Y. D. Tamarkine, “Some general problems of the theory of linear differential equations and expansions of an arbitrary functions in series of fundamental functions”, Math. Z., 27:1 (1928), 1–54 | DOI | MR
[167] H. Volkmer, “Eigenvalue problems of Atkinson, Feller and Krein, and their mutual relationship”, Electron. J. Differ. Equ., 48 (2005) | MR | Zbl
[168] J. Weidmann, Spectral theory of ordinary differential operators, Springer, Berlin, 1987 | MR | Zbl
[169] A. Zettl, “Formally self-adjoint quasi-differential operators”, Rocky Mountain J. Math., 5 (1975), 453–474 | DOI | MR | Zbl