Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



WKB-type approximations for second-order differential equations in $C^\ast $-algebras

Authors: Renato Spigler and Marco Vianello
Journal: Proc. Amer. Math. Soc. 124 (1996), 1763-1771
MSC (1991): Primary 34E20, 34G10, 46L05
MathSciNet review: 1327046
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Rigorous asymptotic approximations of the WKB (or Liouville-Green) type are obtained for a basis of solutions to $y''+[\alpha +g(t)]y=0$ in the framework of $C^\ast $-algebras. Both cases $\alpha >0$ and $\alpha <0$ are included, thus generalizing the classical theory for scalar equations developed by F.W.J. Olver to matrix as well as to infinite-dimensional equations.

References [Enhancements On Off] (What's this?)

  • 1. E. Hille and R.S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Coll. Publ., vol. 31, Amer. Math. Soc., Providence, 1957.MR 19:664d
  • 2. E. Hille, Lectures on ordinary differential equations, Addison-Wesley, Reading, 1969.MR 40:2939
  • 3. F.W.J. Olver, Error bounds for the Liouville-Green (or WKB) approximation, Proc. Cambridge Philos. Soc. 57 (1961), 790-810.MR 24:A313
  • 4. F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. MR 55:8655
  • 5. W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.MR 51:1315
  • 6. D.R. Smith, Decoupling of recessive and nonrecessive solutions for a second-order system, J. Differential Equations 68 (1987), 383-399.MR 88i:34081
  • 7. R. Spigler and M. Vianello, Finite moments perturbations of $y''=0$ in Banach algebras, Proc. Amer. Math. Soc. 119 (1993), 97-103. MR 93k:34144
  • 8. R. Spigler and M. Vianello, Extending L'Hôpital's theorem to B-modules, J. Math. Anal. Appl. 179 (1993), 638-645.MR 94j:46050a
  • 9. M. Vianello, A generalization of L'Hôpital's rule via absolute continuity and Banach modules, Real Anal. Exchange 18 (1993), 557-567.MR 94c:26028

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34E20, 34G10, 46L05

Retrieve articles in all journals with MSC (1991): 34E20, 34G10, 46L05

Additional Information

Renato Spigler
Affiliation: Dipartimento di Matematica, Università di Lecce, Via Provinciale Lecce-Arnesano, P.O. Box 193, 73100, Lecce, Italy

Marco Vianello
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131, Padova, Italy

Keywords: Linear abstract differential equations, WKB (Liouville-Green) approximation, $C^\ast$-algebras
Received by editor(s): November 16, 1994
Additional Notes: This work has been supported, in part, by the MURST Mathematical Analysis funds (“60%" -funds), the GNIM-CNR, and the GNFM-CNR. Hospitality by IMA at the University of Minnesota, Minneapolis, is happily acknowledged by R.S
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society