Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
The Reproducing Kernel Property and Its Space: The Basics Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

17. The Critical Point Infinity Associated with Indefinite Sturm–Liouville Problems

verfasst von : Andreas Fleige

Erschienen in: Operator Theory

Verlag: Springer Basel

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Consider the indefinite Sturm–Liouville problem \(-f^{{\prime\prime}} = \lambda rf\) on [−1, 1] with Dirichlet boundary conditions and with a real weight function r ∈ L 1[−1, 1] changing its sign. The question is studied whether or not the eigenfunctions form a Riesz basis of the Hilbert space L | r | 2[−1, 1] or, equivalently, ∞ is a regular critical point of the associated definitizable operator in the Kreĭn space L r 2[−1, 1]. This question is also related to other subjects of mathematical analysis like half range completeness, interpolation spaces, HELP-type inequalities, regular variation, and Kato’s representation theorems for non-semibounded sesquilinear forms. The eigenvalue problem can be generalized to arbitrary self-adjoint boundary conditions, singular endpoints, higher order, higher dimension, and signed measures. The present paper tries to give an overview over the so far known results in this area.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Indefinite Hamiltonians
Nächstes Kapitel Finite-Dimensional Indefinite Inner Product Spaces and Applications in Numerical Analysis
Literatur
1.
Abasheeva, N.L., Pyatkov, S.G.: Counterexamples in indefinite Sturm–Liouville problems. Sib. Adv. Math. 7(4), 1–8 (1997)MathSciNetMATH
2.
3.
4.
5.
Bennewitz, C.: The HELP inequality in the regular case. Internat. Schriftenreihe Numer. Math. 80, 337–346 (1987)MathSciNetMATH
6.
Bennewitz, C., Brown, B.M., Weikard, R.: Inverse spectral and scattering theory for the half-line left-definite Sturm Liouville problem. SIAM J. Math. Anal. 40(5), 2105–2131 (2009)MathSciNetCrossRefMATH
7.
Binding, P., Ćurgus, B.: A counterexample in Sturm–Liouville completeness theory. Proc. R. Soc. Edinb. A 134, 244–248 (2004)CrossRefMATHMathSciNet
8.
Binding, P., Ćurgus, B.: Riesz bases of root vectors of indefinite Sturm–Liouville problems with eigenparameter dependent boundary conditions, I, II. Oper. Theory Adv. Appl. 163, 75–95 (2006); Integr. Equ. Oper. Theory 63, 473–499 (2009)
9.
Binding, P., Fleige, A.: Conditions for an indefinite Sturm–Liouville Riesz basis property. Oper. Theory Adv. Appl. 198, 87–95 (2009)MathSciNetMATH
10.
Binding, P., Fleige, A.: A review of a Riesz basis property for indefinite Sturm–Liouville problems. Oper. Matrices 5, 735–755 (2011)MathSciNetCrossRefMATH
11.
Binding, P., Hryniv, R.: Full- and partial-range completeness. Oper. Theory Adv. App. 130, 121–133 (2001)MathSciNetMATH
12.
Binding, P., Karabash, I.: Absence of existence and uniqueness for forward–backward parabolic equations on a half-line. Oper. Theory Adv. Appl. 203, 89–98 (2010)MathSciNetMATH
13.
Bingham, N.H., Goldie, C.M., Teugels, J.T.: Regular Variation. Cambridge University Press, Cambridge (1987)CrossRefMATH
14.
Buldygin, V.V., Klesov, O.I., Steinebach, J.S.: On some properties of asymptotic quasi-inverse functions. Theory Probab. Math. Stat. 77, 15–30 (2008)MathSciNetCrossRefMATH
15.
Ćurgus, B.: On the regularity of the critical point infinity of definitizable operators. Integr. Equ. Oper. Theory 8, 462–488 (1985)CrossRefMATHMathSciNet
16.
Ćurgus, B.: Boundary value problems in Krein spaces. Dedicated to the memory of Branko Najman. Glas. Mat. Ser. III 35 (55)(1), 45–5 (2000)MathSciNetMATH
17.
Ćurgus, B.: Orthonormal sets in Krein spaces (in preparation)
18.
Ćurgus, B., Langer, H.: A Krein space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differ. Equ. 79, 31–61 (1989)CrossRefMATHMathSciNet
19.
Ćurgus, B., Najman, B.: A Krein space approach to elliptic eigenvalue problems with indefinite weights. Differ. Integr. Equ. 7, 1241–1252 (1994)MATHMathSciNet
20.
Ćurgus, B., Najman, B.: The operator \((\mathrm{sgn}\,x) \frac{d^{2}} {dx^{2}}\) is similar to a selfadjoint operator in \(L^{2}(\mathbb{R})\). Proc. Am. Math. Soc. 123, 1125–1128 (1995)MATHMathSciNet
21.
Ćurgus, B., Najman, B.: Positive differential operators in Krein space \(L^{2}(\mathbb{R})\). Oper. Theory Adv. Appl. 87, 95–104 (1996)MATHMathSciNet
22.
Ćurgus, B., Najman, B.: Positive differential operators in the Krein space \(L^{2}(\mathbb{R}^{n})\). Oper. Theory Adv. Appl. 106, 113–129 (1998)MATHMathSciNet
23.
Ćurgus, B., Fleige, A., Kostenko, A.: The Riesz basis property of an indefinite Sturm–Liouville problem with non-separated boundary conditions. Integr. Equ. Oper. Theory 77, 533–557 (2013)CrossRefMATHMathSciNet
24.
Daho, K., Langer, H.: Sturm–Liouville operators with an indefinite weight function. Proc. R. Soc. Edinb. Sect. A 87, 161–191 (1977)MathSciNetMATH
25.
Dijksma, A., Langer, H.: Operator theory and ordinary differential operators. In: Lectures on Operator Theory and its Applications, Waterloo, 1994. Fields Institute of Monographs, vol. 3, pp. 73–139. American Mathematical Society, Providence (1996)
26.
Dym, H., McKean, H.P.: Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic, New York/San Francisco/London (1976)MATH
27.
Evans, W.D., Everitt, W.N.: A return to the Hardy-Littlewood integral inequality. Proc. R. Soc. Lond. A 380, 447–486 (1982)MathSciNetCrossRefMATH
28.
Evans, W.D., Everitt, W.N.: HELP inequalities for limit-circle and regular problems. Proc. R. Soc. Lond. A 432, 367–390 (1991)MathSciNetCrossRefMATH
29.
Feller, W.: Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1, 459–504 (1957)MathSciNetMATH
30.
31.
Faierman, M., Langer, H.: Elliptic problems involving an indefinite weight function. Oper. Theory Adv. Appl. 87, 105–127 (1996)MathSciNetMATH
32.
Faierman, M., Roach, G.F.: Full and Half Range Eigenfunction Expansions for an Elliptic Boundary Value Problem Involving an Indefinite Weight. Lecture Notes in Pure and Applied Mathematics, vol. 118, pp. 231–236. Dekker, New York/Basel (1989)
33.
Fleige, A.: The turning point conditionnnn of Beals for indefinite Sturm–Liouville problems. Math. Nachr. 172, 109–112 (1995)MathSciNetCrossRefMATH
34.
Fleige, A.: Spectral Theory of Indefinite Krein–Feller Differential Operators. Mathematical Research, vol. 98. Akademie, Berlin (1996)MATH
35.
Fleige, A.: A counterexample to completeness properties for indefinite Sturm–Liouville problems. Math. Nachr. 190, 123–128 (1998)MathSciNetCrossRefMATH
36.
Fleige, A.: Non-semibounded sesquilinear forms and left-indefinite Sturm–Liouville problems. Integr. Equ. Oper. Theory 33, 20–33 (1999)MathSciNetCrossRefMATH
37.
Fleige, A.: A necessary aspect of the generalized Beals condition for the Riesz Basis property of indefinite Sturm–Liouville problems. Oper. Theory Adv. Appl. 175, 89–94 (2007)MathSciNetCrossRefMATH
38.
Fleige, A.: The Riesz basis property of an indefinite Sturm–Liouville problem with a non odd weight function. Integr. Equ. Oper. Theory 60, 237–246 (2008)MathSciNetCrossRefMATH
39.
Fleige, A.: A failing eigenfunction expansion associated with an indefinite Sturm–Liouville problem. J. Math. Anal. Appl. 389, 932–949 (2012)MathSciNetCrossRefMATH
40.
Fleige, A.: Characterizations of monotone O-regularly varying functions by means of indefinite eigenvalue problems and HELP type inequalities. J. Math. Anal. Appl. 412, 345–359 (2014)MathSciNetCrossRefMATH
41.
Fleige, A., Najman, B.: Nonsingularity of critical points of some differential and difference operators. Oper. Theory Adv. Appl. 102, 85–95 (1998)MathSciNetMATH
42.
Fleige, A., Hassi, S., de Snoo, H.S.V.: A Kreĭn space approach to representation theorems and generalized Friedrichs extensions. Acta Sci. Math. (Szeged) 66, 633–650 (2000)MathSciNetMATH
43.
Fleige, A., Hassi, S., de Snoo, H.S.V., Winkler, H.: Sesquilinear forms corresponding to a non-semibounded Sturm–Liouville operator. Proc. R. Soc. Edinb. A 140, 291–318 (2010)CrossRefMATHMathSciNet
44.
Fleige, A., Hassi, S., de Snoo, H.S.V., Winkler, H.: Non-semibounded closed symmetric forms associated with a generalized Friedrichs extension. Proc. R. Soc. Edinb. A 144, 731–745 (2014)CrossRefMATH
45.
Ganchev, A.H., Greenberg, W., van der Mee, C.V.M.: A class of linear kinetic equations in a Krein space setting. Integr. Equ. Oper. Theory 11, 518–535 (1988)CrossRefMATHMathSciNet
46.
Gohberg, I.C., Kreĭn, M.G.: Introduction to the theory of linear nonselfadjoint operators. In: Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence (1969)
47.
Kac, I.S., Kreĭn, M.G.: On the spectral function of the string. Trans. Am. Math. Soc. Ser. 2(103), 19–102 (1974)MATH
48.
Kaper, H.G., Kwong, M.K., Lekkerkerker, C.G., Zettl, A.: Full- and partial-range eigenfunction expansion for Sturm–Liouville problems with indefinite weights. Proc. R. Soc. Edinb. A 98, 69–88 (1984)MathSciNetCrossRefMATH
49.
Karabash, I.M.: J-selfadjoint ordinary differential operators similar to selfadjoint operators. Methods Funct. Anal. Topol. 6(2), 22–49 (2000)MathSciNetMATH
50.
Karabash, I.M., Kostenko, A.S.: Spectral analysis of differential operators with indefinite weights and a local point interaction. Oper. Theory Adv. Appl. 175, 169–191 (2007)MathSciNetCrossRefMATH
51.
Karabash, I.M., Kostenko, A.S.: Indefinite Sturm–Liouville operators with the singular critical point zero. Proc. R. Soc. Edinb. A 138, 801–820 (2008)MathSciNetCrossRefMATH
52.
Karabash, I.M., Malamud, M.M.: Indefinite Sturm–Liouville operators \((\mathrm{sgn}\,\,x)(-d^{2}/dx^{2} + q(x))\) with finite-zone potentials. Oper. Matrices 1, 301–368 (2007)MathSciNetCrossRefMATH
53.
Karabash, I.M., Kostenko, A.S., Malamud, M.M.: The similarity problem for J-nonnegative Sturm–Liouville operators. J. Differ. Equ. 246, 964–997 (2009)MathSciNetCrossRefMATH
54.
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)MATH
55.
Kostenko, A.: The similarity problem for indefinite Sturm–Liouville operators with periodic coefficients. Oper. Matrices 5, 705–722 (2011)MathSciNetMATH
56.
Kostenko, A.: The similarity problem for indefinite Sturm–Liouville operators and the HELP inequality. Adv. Math. 246, 368–413 (2013)MathSciNetCrossRefMATH
57.
Kostenko, A.: On a necessary aspect for the Riesz basis property for indefinite Sturm–Liouville problems. Math. Nachr. 287, 1710–1732 (2014)MathSciNetCrossRefMATH
58.
Langer, H.: Zur Spektraltheorie verallgemeinerter gewhnlicher Differentialoperatoren zweiter Ordnung mit einer nichtmonotonen Gewichtsfunktion, vol. 14. Universität Jyvskylä, Mathematisches Institut, Bericht (1972)
59.
Langer, H.: Spectral functions of definitizable operators in Krein spaces. In: Butkovic, D., Kraljevic, H., Kurepa, S. (eds.) Functional Analysis. Conf. held at Dubrovnik, November 2–14, 1981. Lecture Notes in Mathematics, vol. 948, pp. 1–46. Springer, Berlin/Heidelberg/New York (1982)
60.
Parfenov, A.I.: On an embedding criterion for interpolation spaces and application to indefinite spectral problems. Sib. Math. J. 44(4), 638–644 (2003)MathSciNetCrossRefMATH
61.
Parfenov, A.I.: The Ćurgus condition in indefinite Sturm–Liouville problems. Sib. Adv. Math. 15(2), 68–103 (2005)MathSciNetMATH
62.
63.
Pyatkov, S.G.: On the solvability of a boundary value problem for a parabolic equation with changing time direction. Soviet Math. Dokl. 32(3), 895–897 (1985)MATH
64.
Pyatkov, S.G.: Some properties of eigenfunctions of linear sheaves. Sibirsk. Mat. Zh. 30(4), 111–124, 218 (1989, Russian); translation in Sib. Math. J. 30(4), 587–597 (1989)
65.
Pyatkov, S.G.: Certain properties of eigenfunctions of linear pencils. Mat. Zametki 51(1), 141–148 (1992, Russian); translation in Math. Notes 51(1–2), 90–95 (1992)
66.
Pyatkov, S.G.: Elliptic eigenvalue problems with an indefinite weight function. Sib. Adv. Math. 4(2), 87–121 (1994)MathSciNetMATH
67.
Pyatkov, S.G.: Riesz completeness of the eigenelements and associated elements of linear selfadjoint pencils. Russian Acad. Sci. Sb. Math. 81(2), 343–361 (1995)MathSciNet
68.
Pyatkov, S.G.: Interpolation of some function spaces and indefinite Sturm–Liouville problems. Oper. Theory Adv. Appl. 102, 179–200 (1998)MathSciNetMATH
69.
70.
Pyatkov, S.G.: Some properties of eigenfunctions and associated functions of indefinite Sturm–Liouville problems. In: Nonclassical Problems of Mathematical Physics, pp. 240–251. Sobolev Institute of Mathematics, Novosibirsk (2005, Russian)
71.
Pyatkov, S.G.: Interpolation of Sobolev spaces and indefinite elliptic spectral problems. Oper. Theory Adv. Appl. 198, 265–290 (2010)MathSciNetMATH
72.
Rogozin, B.A.: A Tauberian theorem for increasing functions of dominated variation. Siberian Math. J. 43(2), 353–356 (2002)MathSciNetCrossRefMATH
73.
Seneta, E.: Regularly Varying Functions. Lecture Notes in Mathematics, vol. 508. Springer, Berlin/Heidelberg/New York (1976)CrossRefMATH
74.
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)MATH
75.
Volkmer, H.: Sturm–Liouville problems with indefinite weights and Everittnns inequality. Proc. R. Soc. Edinb. Sect. A 126, 1097–1112 (1996)MathSciNetCrossRefMATH
Metadaten
Titel
The Critical Point Infinity Associated with Indefinite Sturm–Liouville Problems
verfasst von
Andreas Fleige
Copyright-Jahr
2015
Verlag
Springer Basel
Buch
Operator Theory

Print ISBN: 978-3-0348-0666-4

Electronic ISBN: 978-3-0348-0667-1

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-0348-0667-1_44