Sturm–Liouville oscillation theory for impulsive problems

Yu. V. Pokornyi ; M. B. Zvereva ; S. A. Shabrov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 1, pp. 109-153

Résumé

This paper extends the Sturm–Liouville oscillation theory on the distribution of zeros of eigenfunctions to the case of problems with strong singularities of the coefficients (of $\delta$-function type). For instance, these are problems arising in the study of eigenoscillations of an elastic continuum with concentrated masses and localized interactions with the surrounding medium. The extension of the standard description of the problem is carried out by replacing the usual form of the ordinary differential equation $$ -(pu')'+qu=\lambda mu $$ by the substantially more general form $$ -(pu')(x)+(pu')(0)+\int_0^xu\,dQ=\lambda\int_0^xu\,dM $$ with absolutely continuous solutions whose derivatives, as well as the coefficients $p$, $Q$, $M$, belong to $\operatorname{BV}[0,l]$. The integral is understood in the Stieltjes sense.

Bibliographie

[1] C. Sturm, “Mémoire sur une classe d'équations à différences partielles”, J. Math. Pures Appl., 1 (1836), 373–444

[2] F. V. Atkinson, Discrete and continuous boundary problems, Math. Sci. Eng., 8, Academic Press, New York–London, 1964 | MR | Zbl | Zbl

[3] E. L. Ince, Integration of ordinary differential equations, Oliver and Boyd, Edinburgh, 1939 | MR | Zbl

[4] B. M. Levitan, Razlozhenie po sobstvennym funktsiyam differentsialnykh uravnenii vtorogo poryadka, GITTL, M.–L., 1950 | MR

[5] I. G. Petrovskii, Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Uchebnoe posobie, MGU, M., 1984 | MR | Zbl

[6] E. A. Coddington, N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York–Toronto–London, 1955 | MR | Zbl

[7] G. Sansone, Equazioni differenziali nel campo reale, vol. I, Zanichelli, Bologna, 1948 | MR | MR | Zbl

[8] F. Tricomi, Equazioni differenziali, Einaudi, Torino, 1948 | MR | Zbl

[9] N. Dunford, J. T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Wiley, New York–London, 1963 | MR | MR | Zbl

[10] F. R. Gantmakher, M. G. Krein, Ostsillyatsionnye matritsy i yadra i malye kolebaniya mekhanicheskikh sistem, GITTL, M.–L., 1950 | MR

[11] F. R. Gantmakher, “O nesimmetricheskikh yadrakh Kelloga”, Dokl. AN SSSR, 1 (1936), 3–5 | Zbl

[12] A. Yu. Levin, G. D. Stepanov, “One-dimensional boundary value problems with operators that do not lower the number of sign changes. I”, Siberian Math. J., 17:3 (1976), 466–482 | DOI | DOI | MR | MR | Zbl | Zbl

[13] Siberian Math. J., 17:4 (1976), 612–625 | DOI | DOI | MR | MR | Zbl | Zbl

[14] Yu. V. Pokornyi, “A nonclassical de la Vallée-Poussin problem”, Differ. Equ., 14:6 (1978), 725–732 | MR | Zbl | Zbl

[15] A. V. Borovskikh, Yu. V. Pokornyi, “Chebyshev–Haar systems in the theory of discontinuous Kellogg kernels”, Russian Math. Surveys, 49:3 (1994), 1–42 | DOI | MR | Zbl

[16] Yu. V. Pokornyi, K. P. Lazarev, “Some oscillation theorems for many-point problems”, Differ. Equ., 23:4 (1987), 452–462 | MR | Zbl

[17] V. Ya. Derr, “On the generalized de la Vallée-Poussin problem”, Differ. Equ., 23:11 (1987), 1254–1263 | MR | Zbl | Zbl

[18] A. F. Fillipov, Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR | Zbl

[19] J. Kurzweil, “Generalized ordinary differential equations”, Czechoslovak Math. J., 8(83) (1958), 360–388 | MR | Zbl

[20] A. Halanay, D. Wexler, Teoria calitativă a sistemelor cu impulsuri, Editura Academiei Republicii Socialiste Romania, Bucharest, 1968 | MR | MR | Zbl | Zbl

[21] P. Antosik, J. Mikusinski, R. Sikorski, Theory of distributions. The sequential approach, Elsevier, Amsterdam; PWN–Polish Scientific Publishers, Warsaw, 1973 | MR | MR | Zbl

[22] A. Yu. Levin, “Voprosy teorii obyknovennogo lineinogo differentsialnogo uravneniya. II”, Vestn. YarGU, 1974, no. 8, 122–144 | MR

[23] L. Young, Lectures on the calculus of variations and optimal control theory, Saunders, Philadelphia–London–Toronto, 1969 | MR | MR | Zbl

[24] E. L. Tonkov, “K voprosu o neostsillyatsii lineinoi sistemy”, Nelineinye kolebaniya i teoriya upravleniya, no. 4, Udmurt. gos. un-t, Izhevsk, 1982, 62–74 | MR

[25] S. T. Zavalischin, A. N. Sesekin, Impulsnye protsessy: modeli i prilozheniya, Nauka, M., 1991 | MR | Zbl

[26] V. P. Maksimov, “O nekotorykh obobscheniyakh obyknovennykh differentsialnykh uravnenii, kraevykh zadach i ikh prilozheniyakh k zadacham ekonomicheskoi dinamiki”, Vest. PermGU, 4 (1997), 103–120

[27] A. M. Savchuk, A. A. Shkalikov, “Sturm–Liouville operators with singular potentials”, Math. Notes, 66:6 (1999), 741–753 | DOI | MR | Zbl

[28] V. Ya. Derr, D. M. Kinzebulatov, “Differentsialnye uravneniya s obobschennymi funktsiyami, dopuskayuschimi umnozhenie na razryvnye funktsii”, Vestn. Udmurt. un-ta, 2005, no. 1, 35–58

[29] V. I. Rodionov, “Prisoedinennyi integral Rimana–Stiltesa v algebre preryvistykh funktsii”, Izv. In-ta matem. i inform. UGU, Izhevsk, 2005, 3–78

[30] V. A. Dykhta, O. N. Samsonyuk, Optimalnoe impulsnoe upravlenie s prilozheniyami, Fizmatlit, M., 2000 | MR | Zbl

[31] Yu. V. Egorov, Linear differential equations of principal type, Contemp. Soviet Math., Consultants Bureau, New York, 1986 | MR | MR | Zbl

[32] A. D. Myshkis, “Solutions of a linear homogeneous binomial second-order differential inequality with a distribution as a coefficient”, Differ. Equ., 32:5 (1996), 619–623 | MR | Zbl

[33] Yu. V. Pokornyi, “The Stieltjes integral and derivatives with respect to the measure in ordinary differential equations”, Dokl. Math., 59:1 (1999), 34–37 | MR | Zbl

[34] Yu. V. Pokornyi, “The Stieltjes integral and derivatives with respect to the measure in ordinary differential equations”, Dokl. Math., 65:2 (2002), 262–265 | MR | Zbl

[35] Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov, “Ob odnom klasse obobschennykh zadach Shturma–Liuvillya s razryvnymi resheniyami”, Differentsialnye uravneniya i smezhnye voprosy, Mezhdunarod. konf., posvyasch. 103-letiyu so dnya rozhd. I. G. Petrovskogo (Moskva, 2004), Sbornik tezisov, 166–167

[36] Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov, “O zadache Shturma–Liuvillya dlya razryvnoi struny”, Izv. VUZov Severo-Kavkaz. regiona. Estestv. nauki. Matem. i mekh. sploshnoi sredy, Spetsvypusk, 2004, 186–191 | Zbl

[37] Yu. V. Pokornyi, S. A. Shabrov, “Lineinye differentsialnye uravneniya vtorogo poryadka s obobschennymi koeffitsientami”, Tr. matem. f-ta VGU (nov. ser.), 4 (1999), 84–96

[38] Yu. V. Pokornyi, S. A. Shabrov, “Toward a Sturm–Liouville theory for an equation with generalized coefficients”, J. Math. Sci., 119:6 (2004), 769–787 | DOI | MR | Zbl

[39] Yu. V. Pokornyi, O. M. Penkin, V. L. Pryadiev i dr. (red.), Differentsialnye uravneniya na geometricheskikh grafakh, Fizmatlit, M., 2004 | Zbl

[40] I. S. Kac, M. G. Krein, “On the spectral functions of the string”, Nine papers in analysis, Amer. Math. Soc. Transl., Ser. II, 103, Amer. Math. Soc., Providence, R.I., 1974, 19–102 | MR | MR | Zbl

[41] W. Feller, “Generalized second order differential operators and their lateral conditions”, Illinois J. Math., 1 (1957), 459–504 | MR | Zbl

[42] Yu. V. Pokornyi, V. L. Pryadiev, “Some problems of the qualitative Sturm–Liouville theory on a spatial network”, Russian Math. Surveys, 59:3 (2004), 515–552 | DOI | MR | Zbl

[43] W. Rudin, Principles of mathematical analysis, 2nd ed., New York–San Francisco–Toronto-London, McGraw-Hill; Tokyo, Kogakusha, 1964 | MR | Zbl | Zbl

[44] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR | Zbl

[45] S. Saks, Theory of the integral, 2, revised edit., G. E. Stechert Co. VI, New York, 1937 | MR | Zbl

[46] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968 | MR | Zbl

[47] M. I. Dyachenko, P. L. Ulyanov, Mera i integral, Faktorial, M., 2002

[48] N. Dunford, J. T. Schwartz, Linear Operators. I. General theory, Pure Appl. Math., 6, Interscience, New York–London, 1958 | MR | MR | Zbl

[49] F. Riesz, B. Sz.-Nagy, Leçons d'analyse fonctionnelle, Académie des Sciences de Hongrie, Budapest, 1952 | MR | MR | Zbl

[50] E. F. Beckenbach, R. Bellman, Inequalities, Ergeb. Math. Grenzgeb., 30, Springer, Berlin–Göttingen–Heidelberg, 1961 | MR | MR | Zbl | Zbl

[51] R. Courant, D. Hilbert, Methoden der mathematischen Physik. B. I, Interscience, New York, 1943 | MR | MR | Zbl

[52] E. Kamke, Differentialgleichungen. Losungsmethoden und Losungen, Akademische Verlagsgesellschaft, Leipzig, 1944 | MR | MR | Zbl | Zbl

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

[54] Yu. V. Pokornyi, A. V. Borovskikh, “On defects of axiomatics of Green's function”, Dokl. Math., 65:3 (2002), 399–403 | MR | MR | Zbl

[55] M. A. Krasnosel'skij, Je. A. Lifshits, A. V. Sobolev, Positive linear systems. The method of positive operators, Sigma Ser. Appl. Math., 5, Heldermann-Verlag, Berlin, 1989 | MR | MR | Zbl | Zbl

[56] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii i deistviya nad nimi, Fizmatlit, M., 1959 ; I. M. Gel'fand, G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York, London, 1964 ; Generalized functions. Vol. 2: Spaces of fundamental and generalized functions, 1968 ; Generalized functions. Vol. 3. Theory of differential equations, 1967 | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[57] I. C. Gohberg, M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, R.I., 1969 | MR | MR | Zbl | Zbl

[58] B. M. Levitan, I. S. Sargsjan, Introduction to spectral theory: Selfadjoint ordinary differential operators, Transl. Math. Monogr., 39, Amer. Math. Soc., Providence, R.I., 1975 | MR | MR | Zbl | Zbl

[59] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag, New York, 1966 | MR | MR | Zbl | Zbl

[60] V. A. Marchenko, Sturm–Liouville operators and applications, Oper. Theory Adv. Appl., 22, Birkhäuser, Basel–Boston–Stuttgart, 1986 | MR | MR | Zbl | Zbl

[61] M. Reed, B. Simon, Methods of modern mathematical physics. II: Harmonic analysis. Selfadjointness, Academic Press, New York–London, 1975 | MR | MR | Zbl

[62] B. M. Levitan, Inverse Sturm–Liouville problems, VNU Science Press, Utrecht, 1987 | MR | MR | Zbl | Zbl

[63] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable models in quantum mechanics, Texts and Monogr. Phys., Springer-Verlag, New York, 1988 | MR | MR | Zbl

[64] I. M. Gelfand, B. M. Levitan, “Ob opredelenii differentsialnogo uravneniya po ego spektralnoi funktsii”, Dokl. AN SSSR, 77 (1951), 557–560 | MR | Zbl

[65] B. M. Levitan, “Vychislenie regulyarizovannogo sleda dlya operatora Shturma–Liuvillya”, UMN, 19:1 (1964), 161–165 | MR | Zbl

[66] V. A. Sadovnichii, “O tozhdestvakh dlya sobstvennykh znachenii sistemy Diraka i nekotorykh drugikh sistem vysshego poryadka”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 22:3 (1967), 37–47 | MR | Zbl

[67] M. V. Keldysh, “On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators”, Russian Math. Surveys, 26:4 (1971), 15–44 | DOI | MR | Zbl

[68] V. F. Lazutkin, T. F. Pankratova, “Asymptotics of the width of gaps in the spectrum of the Sturm–Liouville operator with a periodic potential”, Soviet Dokl. Math., 15 (1974), 649–653 | MR | Zbl

[69] P. Kurasov, “On the Coulomb potentials in one dimension”, J. Phys. A, 29:8 (1996), 1767–1771 | DOI | MR | Zbl

[70] G. V. Radzievskii, “Asymptotics of the eigenvalues of a regular boundary value problem”, Ukrainian Math. J., 48:4 (1997), 537–575 | DOI | MR | Zbl

[71] P. Djakov, B. Mityagin, “Spectra of 1-D periodic Dirac operators and smoothness of potentials”, C. R. Math. Acad. Sci. Soc. R. Can., 25:4 (2003), 121–125 | MR | Zbl

[72] R. O. Hryniv, Ya. V. Mykytyuk, “Inverse spectral problems for Sturm–Liouville operators with singular potentials”, Inverse Problems, 19:3 (2003), 665–684 | DOI | MR | Zbl

[73] M. G. Gasymov, “Spectral analysis of a class of second-order non-self-adjoint differential operatorsm”, Funct. Anal. Appl., 14:1 (1980), 11–15 | DOI | MR | Zbl

[74] V. A. Mikhailets, “A discreteness criterion for the spectrum of a one-dimensional Schrödinger operator with $\delta$-interactions”, Funct. Anal. Appl., 28:4 (1994), 290–292 | DOI | MR | Zbl

[75] V. A. Vinokurov, V. A. Sadovnichij, “The eigenvalue and trace of the Sturm–Liouville operator as differentiable functions of a summable potential”, Dokl. Math., 59:2 (1999), 220–222 | MR | Zbl

[76] M. I. Neiman-zade, A. A. Shkalikov, “Schrödinger operators with singular potentials from the space of multiplicators”, Math. Notes, 66:5 (1999), 599–607 | DOI | MR | Zbl

[77] E. Korotyaev, “Characterization of the spectrum of Schrödinger operator with periodic distributions”, Int. Math. Res. Not., 2003, no. 37, 2019–2031 | DOI | MR | Zbl

[78] B. S. Mityagin, “Skhodimost razlozhenii po sobstvennym funktsiyam operatora Diraka”, Dokl. RAN, 393:4 (2003), 456–459 | MR

[79] A. P. Khromov, “Integral operators with kernels that are discontinuous on broken lines”, Sb. Math., 197:11 (2006), 1669–1696 | DOI

[80] V. A. Mikhailets, “The structure of the continuous spectrum of a one-dimensional Schrödinger operator with point interactions”, Funct. Anal. Appl., 30:2 (1996), 144–146 | DOI | MR | Zbl

[81] A. M. Savchuk, A. A. Shkalikov, “Trace formula for Sturm–Liouville operators with singular potentials”, Math. Notes, 69:3–4 (2001), 387–400 | DOI | MR | Zbl

[82] A. M. Savchuk, “On the eigenvalues and eigenfunctions of the Sturm–Liouville operator with a singular potential”, Math. Notes, 69:1–2 (2001), 245–252 | DOI | MR | Zbl

[83] J.-G. Bak, A. A. Shkalikov, “Multipliers in dual Sobolev spaces and Schrödinger operators with distribution potentials”, Math. Notes, 71:5–6 (2002), 587–594 | DOI | MR | Zbl

[84] A. M. Savchuk, A. A. Shkalikov, “Sturm–Liouville operators with distribution potentials”, Trans. Moscow Math. Soc., 2003, 143–192 | MR | Zbl

[85] A. M. Savchuk, A. A. Shkalikov, “On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces”, Math. Notes, 80:5–6 (2006), 814–832 | DOI | MR | Zbl

[86] E. A. Shiryaev, A. A. Shkalikov, “Regular and completely regular differential operators”, Math. Notes, 81:3–4 (2007), 566–570 | DOI | MR | Zbl

[87] P. Djakov, B. Mityagin, “Asymptotics of instability zones of Hill operators with a two term potential”, C. R. Math. Acad. Sci. Paris, 339:5 (2004), 351–354 | DOI | MR | Zbl

[88] P. Djakov, B. Mityagin, “Instability zones of a periodic $1D$ Dirac operator and smoothness of its potential”, Comm. Math. Phys., 259:1 (2005), 139–183 | DOI | MR | Zbl

[89] P. Djakov, B. S. Mityagin, “Instability zones of periodic $1$-dimensional Schrödinger and Dirac operators”, Russian Math. Surveys, 61:4 (2007), 663–766 | DOI | MR | Zbl

[90] R. O. Hryniv, Ya. V. Mykytyuk, “Inverse spectral problems for Sturm–Liouville operators with singular potentials. IV. Potentials in Sobolev space scale”, Proc. Edinb. Math. Soc. (2), 49:2 (2006), 309–329 | DOI | MR | Zbl