article

Free access

Using the SLEDGE package on Sturm-Liouville problems having nonempty essential spectra

Editor: Ronald Boisvert Authors:

Michael S. P. Eastham,

Charles T. Fulton,

Steven PruessAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 22, Issue 4

Pages 423 - 446

https://doi.org/10.1145/235815.235819

Published: 01 December 1996 Publication History

PDF eReader

Abstract

We describe the performance of the Sturm-Liouville software package SLEDGE on a variety of problems having continuous spectra. The code's output is shown to be in good accord with a wide range of known theoretical results.

References

[1]

ARONSZAJN, N. 1957. On a problem of Weyl in the theory of singular Sturm-Liouville equations. Am. J. Math. 79, 597-610.

Google Scholar

[2]

BAILEY, P., GARBOW, B., KAPER, H., AND ZETTL, A. 1991a. Eigenvalue and eigenfunction computations for Sturm-Liouville problems. ACM Trans. Math. Softw. 17, 491-499.

Crossref

Google Scholar

[3]

BAILEY, P., GARBOW, B., KAPER, H., AND ZETTL, A. 1991b. Algorithm 700: A FORTRAN software package for Sturm-Liouville problems. ACM Trans. Math. Softw. 17, 500-501.

Crossref

Google Scholar

[4]

BAILEY, P., GORDON, M., AND SttAMPINE, L. 1978. Automatic solution of the Sturm-Liouville problem. ACM Trans. Math. Softw. 4, 193-208.

Crossref

Google Scholar

[5]

CODDINGTON, E. AND LEVINSON, N. 1955. Theory of Ordinary Differential Equations. McGraw-Hill, New York.

Google Scholar

[6]

EASTItAM, M. S.P. 1973. The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh, U.K.

Google Scholar

[7]

EASTItAM, M. S.P. 1977. Sturm-Liouville equations and purely continuous spectra. Nieuw. Arc. Wisk. 25, 169-181.

Google Scholar

[8]

EASTItAM, M. S.P. 1989. The Asymptotic Solution of Lineal' Differential Systems: Applications of the Levinson Theorem. London Mathematical Society Monographs, vol. 4. Clarendon Press, Oxford, U.K.

Google Scholar

[9]

EASTtIAM, M. S.P. 1991. The number of resonant states in perturbed harmonic oscillation. Q. d. Math. 42, 49-55.

Google Scholar

[10]

EASTHAM, M. S. P. AND KALF, H. 1982. Schroedinger-type operators with continuous spectra. In Research Notes in Mathematics. Vol. 65. Pitman Advanced Publishing, London.

Google Scholar

[11]

EASTHAM, M. S. P. AND McLEoD, J.B. 1977. The existence of eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems, d. Math. Anal. Appl. 188, 297-314.

Google Scholar

[12]

FULTON, C. AND PRUESS, S. 1995. The computation of spectral density functions for singular Sturm-Liouville problems involving simple, continuous spectra. Colorado School of Mines, Golden, Colo. Available via www.mines.edu/fshome/spruess/recent.html or ftp.lyapunov. mine s.edu/pub/sprues s/sledge paper s/.

Google Scholar

[13]

FULTON, C. AND PRUESS, S. 1994. Numerical approximation of singular spectral functions arising from the Fourier-Jacobi problem on a half-line with continuous spectra. In Fourier Analysis: Analytic and Geometric Aspects (Proceedings of the 6th International Workshop in Analysis and Its Applications (IWAA)), W. Bray, P. Milojevic, and P. Stanojevic, Eds. Marcel Dekker, New York, 149-169.

Google Scholar

[14]

FULTON, C., PRUESS, S., AND XIE, Y. 1997. The automatic classification of Sturm-Liouville problems. J. Appl. Math. Cornput. To be published.

Google Scholar

[15]

GELFAND, I. AND LEVITAN, B. 1955. On the determination of a differential equation from its spectral function. Am. Math. Soc. Transl. 2, 253-304.

Google Scholar

[16]

GLAZMAN, I.M. 1965. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators. S. Monson, Jerusalem.

Google Scholar

[17]

HARTMAN, P. AND WINTNER, A. 1949. A separation theorem for continuous spectra. Am. J. Math. 71, 650-662.

Google Scholar

[18]

INCE, E. 1931. Tables of the elliptic-cylinder functions. Proc. Roy. Soc. Edinburgh 52, 355-423.

Google Scholar

[19]

LEVITAN, B. AND GASYMOV, M. 1964. Determination of a differential equation by two of its spectra. Russ. Math. Surv. 19, 1-63.

Google Scholar

[20]

NAG. 1988. NAG FORTRAN Library (Mark 13 documentation). Vol. 2. Numerical Algorithms Group, Ltd., Oxford, U.K.

Google Scholar

[21]

MARLETTA, M. AND PRYCE, J.D. 1992. Automatic solution of Sturm-Liouville problems using the Pruess method. J. Cornput. Appl. Math. 39, 57-78.

Crossref

Google Scholar

[22]

PRUESS, S. AND FULTON, C. 1993. Mathematical software for Sturm-Liouville problems. ACM Trans. Math. Softw. 19, 360-376.

Crossref

Google Scholar

[23]

PRUESS, S., FULTON, C., AND XIE, Y. 1995. An asymptotic numerical method for a class of singular Sturm-Liouville problems. SIAM J. Numer. Anal. 32, 1658-1676.

Crossref

Google Scholar

[24]

PRUESS, S., XIE, Y., AND FULTON, C. 1994. Performance of the Sturm-Liouville software package SLEDGE. Tech Rep. MCS-91-19, Dept. of Mathematics and Computer Science, Colorado School of Mines, Golden, Colo. Revised Sept. 1994.

Google Scholar

[25]

PRYCE, J. AND MARLETTA, M. 1991. A new multi-purpose software package for SchrSdinger and Sturm-Liouville computations. Cornput. Phys. Cornmun. 62, 42-50.

Google Scholar

[26]

THURLOW, C. 1979a. A generalization of the inverse spectral theorem of Levitan and Gasymov. Proc. Roy. Soc. Edinburgh 84A, 187-196.

Google Scholar

[27]

THURLOW, C. 1979b. The point continuous spectrum of second order ordinary differential operators. Proc. Roy. Soc. Edinburgh 84A, 197-211.

Google Scholar

[28]

TITCttMARStt, E.C. 1962. Eigenfunction Expansions Associated with Second Order Differential Equations, Part I. 2nd ed. Oxford University Press, New York.

Google Scholar

[29]

VON NEUMANN, J. AND WIGN~R, E. 1929. 'IZber merkwfirdige, diskrete eigenwerte. Phys. Z. 30, 465-476.

Google Scholar

[30]

WINTNER, A. 1946. The adiabatic linear oscillator. Am. J. Math. 68, 385-397.

Google Scholar

[31]

WINTNER, A. 1947. Asymptotic integrations of the adiabatic oscillator. Am. J. Math. 69, 251-272.

Google Scholar

Cited By

View all

Lehman ENegulescu C(2025)Fokker-Planck equation for energetic particles. The $ \kappa $-distribution functionKinetic and Related Models10.3934/krm.2025004(0-0)Online publication date: 2025
https://doi.org/10.3934/krm.2025004
Baeyens TVan Daele M(2021)The fast and accurate computation of eigenvalues and eigenfunctions of time-independent one-dimensional Schrödinger equationsComputer Physics Communications10.1016/j.cpc.2020.107568258(107568)Online publication date: Jan-2021
https://doi.org/10.1016/j.cpc.2020.107568
ZHANG YHUANG ZZHANG X(2010)Computing Eigenvalues of Left-Definite Sturm-Liouville ProblemsActa Analysis Functionalis Applicata10.3724/SP.J.1160.2010.0011012:2(110-114)Online publication date: 19-Sep-2010
https://doi.org/10.3724/SP.J.1160.2010.00110
Show More Cited By

Index Terms

Using the SLEDGE package on Sturm-Liouville problems having nonempty essential spectra
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
  2. Mathematical software

Recommendations

The computation of spectral density functions for singular Sturm-Liouville problems involving simple continuous spectra

The software package SLEDGE has as one of its options the estimation of spectral density functions p(t) for a wide class of singular Strurm-Liouville problems. In this article the underlaying theory and implementation issues are discussed. Several ...
Mathematical software for Sturm-Liouville problems

Software is described for the Sturm-Liouville eigenproblem. Eigenvalues, eigenfunctions, and spectral density functions can be estimated with global error control. The method of approximating the coefficients forms the mathematical basis. The underlying ...
Matrix representations of Sturm-Liouville problems with transmission conditions

We identify a class of Sturm-Liouville equations with transmission conditions such that any Sturm-Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 22, Issue 4

Dec. 1996

116 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/235815

Editor:
Ronald Boisvert
National Institute of Standards and Technology, Guithersburg, MD

Issue’s Table of Contents

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 December 1996

Published in TOMS Volume 22, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
504
Total Downloads

Downloads (Last 12 months)65
Downloads (Last 6 weeks)7

Reflects downloads up to 05 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Lehman ENegulescu C(2025)Fokker-Planck equation for energetic particles. The $ \kappa $-distribution functionKinetic and Related Models10.3934/krm.2025004(0-0)Online publication date: 2025
https://doi.org/10.3934/krm.2025004
Baeyens TVan Daele M(2021)The fast and accurate computation of eigenvalues and eigenfunctions of time-independent one-dimensional Schrödinger equationsComputer Physics Communications10.1016/j.cpc.2020.107568258(107568)Online publication date: Jan-2021
https://doi.org/10.1016/j.cpc.2020.107568
ZHANG YHUANG ZZHANG X(2010)Computing Eigenvalues of Left-Definite Sturm-Liouville ProblemsActa Analysis Functionalis Applicata10.3724/SP.J.1160.2010.0011012:2(110-114)Online publication date: 19-Sep-2010
https://doi.org/10.3724/SP.J.1160.2010.00110
Fulton CPearson DPruess S(2005)Computing the spectral function for singular Sturm-Liouville problemsJournal of Computational and Applied Mathematics10.5555/1056329.1716603176:1(131-162)Online publication date: 1-Apr-2005
https://dl.acm.org/doi/10.5555/1056329.1716603
Fulton CPearson DPruess S(2005)Computing the spectral function for singular Sturm–Liouville problemsJournal of Computational and Applied Mathematics10.1016/j.cam.2004.07.006176:1(131-162)Online publication date: Apr-2005
https://doi.org/10.1016/j.cam.2004.07.006
Brown BEvans W(2002)Prof. Michael Eastham LaudatumJournal of Computational and Applied Mathematics10.5555/638714.638716148:1(.11-.13)Online publication date: 1-Nov-2002
https://dl.acm.org/doi/10.5555/638714.638716
Brown BEvans W(2002)Prof. Michael Eastham LaudatumJournal of Computational and Applied Mathematics10.1016/S0377-0427(02)00756-2148:1(xi-xiii)Online publication date: Nov-2002
https://doi.org/10.1016/S0377-0427(02)00756-2
Eastham M(1998)On the location of spectral concentration for Sturm–Liouville problems with rapidly decaying potentialMathematika10.1112/S002557930001401745:1(25-36)Online publication date: Jun-1998
https://doi.org/10.1112/S0025579300014017
Brown BEastham MMcCormack D(1998)Absolute continuity and spectral concentration for slowly decaying potentialsJournal of Computational and Applied Mathematics10.1016/S0377-0427(98)00087-994:2(181-197)Online publication date: 3-Aug-1998
https://dl.acm.org/doi/10.1016/S0377-0427%2898%2900087-9
Brown BEastham MMcCormack D(1997)Spectral concentration and rapidly decaying potentialsJournal of Computational and Applied Mathematics10.1016/S0377-0427(97)00072-181:2(333-348)Online publication date: 8-Jul-1997
https://dl.acm.org/doi/10.1016/S0377-0427%2897%2900072-1

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Abstract

References

Cited By

Index Terms

Recommendations

The computation of spectral density functions for singular Sturm-Liouville problems involving simple continuous spectra

Mathematical software for Sturm-Liouville problems

Matrix representations of Sturm-Liouville problems with transmission conditions

Comments

Information

Published In

Publisher

Publication History

Permissions

Check for updates

Author Tags

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

PDF

eReader

Login options

Full Access

Share

Share this Publication link

Share on social media

Affiliations