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.
KeywordsLiouville Operator Completeness Relation Positive Real Axis Riemann Sheet Adjoint Boundary Condition
Unable to display preview. Download preview PDF.
- Coddington, E.A. and Levinson, N. (1984). Theory of Ordinary Differential Equations, Malabar: Robert E. Krieger.Google Scholar
- Dennery, P. and Krzywicki, A. (1995). Mathematics for Physicists, New York: Dover.Google Scholar
- Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical Functions, New York: Dover.Google Scholar
- Stakgold, I. (1999). Green’s Functions and Boundary Value Problems, 2nd ed., New York: Wiley.Google Scholar
- Krall, A.M. (1973). Linear Methods in Applied Analysis, Reading, MA: Addison-Wesley.Google Scholar
- Mrozowski, M. (1997). Guided Electromagnetic Waves, Properties and Analysis, Somerset, England: Research Studies Press.Google Scholar
- Gohberg, I. and Goldberg, S. (1980) Basic Operator Theory, Boston: Birkhäuser.Google Scholar