Sturm—Liouville Operators

  • George W. Hanson
  • Alexander B. Yakovlev


Sturm—Liouville equations arise in many applications of electromagnetics, including in the formulation of waveguiding problems using scalar potentials, and using scalar components of vector fields and potentials. Sturm— Liouville equations are also encountered in separation-of-variables solutions to Laplace and Helmholtz equations, making a connection with certain special functions and classical polynomials. Spectral theory of the Sturm— Liouville operator is well developed and useful and is intertwined with the spectral theory of compact, self-adjoint operators through the inverse and resolvent operators. Accordingly, eigenfunctions of the regular Sturm— Liouville operator are found to be complete in certain weighted spaces of Lebesgue square-integrable functions, and generalizations to accommodate a continuous spectrum in the singular case are possible.


Liouville Operator Completeness Relation Positive Real Axis Riemann Sheet Adjoint Boundary Condition 
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  1. [1]
    Dudley, D.G. (1994). Mathematical Foundations for Electromagnetic Theory, New York: IEEE Press.CrossRefGoogle Scholar
  2. [2]
    Naimark, M.A. (1967). Linear Differential Operators, Part I, New York: Frederick Ungar.zbMATHGoogle Scholar
  3. [3]
    Mikhlin, S. G. (1970). Mathematical Physics, An Advanced Course, Amsterdam: North-Holland.zbMATHGoogle Scholar
  4. [4]
    Naylor, A.W. and Sell, G.R. (1982). Linear Operator Theory in Engineering and Science, 2nd ed., New York: Springer-Verlag.zbMATHCrossRefGoogle Scholar
  5. [5]
    Debnath, L. and Mikusiński, P. (1999). Introduction to Hilbert Spaces with Applications, San Diego: Academic Press.zbMATHGoogle Scholar
  6. [6]
    Kreyszig, E. (1993). Advanced Engineering Mathematics, 7th ed., New York: Wiley.zbMATHGoogle Scholar
  7. [7]
    Stakgold, I. (1967). Boundary Value Problems of Mathematical Physics, Vol.1, New York: Macmillan.zbMATHGoogle Scholar
  8. [8]
    Coddington, E.A. and Levinson, N. (1984). Theory of Ordinary Differential Equations, Malabar: Robert E. Krieger.Google Scholar
  9. [9]
    Dennery, P. and Krzywicki, A. (1995). Mathematics for Physicists, New York: Dover.Google Scholar
  10. [10]
    Friedman, B. (1956). Principles and Techniques of Applied Mathematics, New York: Dover.zbMATHGoogle Scholar
  11. [11]
    Gohberg, I.C. and Krein, M.G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators, Providence, RI: American Mathematical Society.zbMATHGoogle Scholar
  12. [12]
    Ramm, A. G. (1982). Mathematical foundations of the singularity and eigenmode expansion method J. Math. Anal. Appl., Vol. 86, pp. 562–591 (see also Mathematics Notes, Note 68, Phillips Laboratory Note Series, Kirtland AFB, Dec. 1980).MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Jackson, J.D. (1975). Classical Electrodynamics, 2nd ed., New York: John Wiley.zbMATHGoogle Scholar
  14. [14]
    Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical Functions, New York: Dover.Google Scholar
  15. [15]
    Stakgold, I. (1999). Green’s Functions and Boundary Value Problems, 2nd ed., New York: Wiley.Google Scholar
  16. [16]
    Krall, A.M. (1973). Linear Methods in Applied Analysis, Reading, MA: Addison-Wesley.Google Scholar
  17. [17]
    Mrozowski, M. (1997). Guided Electromagnetic Waves, Properties and Analysis, Somerset, England: Research Studies Press.Google Scholar
  18. [18]
    Gohberg, I. and Goldberg, S. (1980) Basic Operator Theory, Boston: Birkhäuser.Google Scholar
  19. [19]
    Morse, P.M. and Feshbach, H. (1953). Methods of Theoretical Physics, Vol. I, New York: McGraw-Hill.zbMATHGoogle Scholar
  20. [20]
    Jones, D.S. (1994). Methods in Electromagnetic Wave Propagation, 2nd ed., New York: IEEE Press.CrossRefGoogle Scholar
  21. [21]
    Lang, P. and Locker, J. (1989). Spectral theory of two-point differential operators determined by -D2. I. Spectral Properties, J. Math. Anal. Appl., Vol. 141, pp. 538–558, Aug.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Lang, P. and Locker, J. (1990). Spectral theory of two-point differential operators determined by -D2. I. Analysis of Cases, J. Math. Anal. Appl., Vol. 146, pp. 148–191, Feb.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Locker, J. (2000). Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators, Providence, RI: American Mathematical Society.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • George W. Hanson
    • 1
  • Alexander B. Yakovlev
    • 2
  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of WisconsinMilwaukeeUSA
  2. 2.Department of Electrical EngineeringUniversity of MississippiUniversityUSA

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