# Sturm-Liouville Theory. The Sturm-Liouville Eigenvalue Problem

## Form of the S-L Equation

Many eigenvalue problems in quantum mechanics as well as classical physics fit into the class of DE called Sturm-Liouville equations:

Equ (1)

where y(x) is the quantum mechanical wave function or other physical quantity, f(x), g(x), and h(x) are real functions of x determined by the nature of the system of interest, and l is a parameter (perhaps the energy of the system) that adopts characteristic or eigen-values to satisfy boundary conditions on y(x) at end points of an interval a<x<b.

Exercise and Example

Show that the vibrating string problem is an example of the S-L eigenvalue problem. do this by deducing the form of the functions f(x), g(x) and h(x) in this case. Other examples of Sturm-Liouville problems in physics abound.

An important example is the Legendre DE that gives rise to Legendre polynomials.

## Solutions of S-L Problems

We now describe solutions of the Sturm-Liouville problem in those cases with h(x) = 1, and boundary conditions of the form y(a)=y(b)=0. These limitations are appropriate for most quantum mechanics problems as well as many classical problems. Solutions for these special S-L problems are described in several sources (e.g., H. Margenau & G. M. Murphy, The Mathematics of Physics and Chemistry [D. van Nostrand, 1956]). The properties of the general solutions are summarized in the following few paragraphs.

If f(x) and h(x) are positive in the interval a<x<b then the S-L equation is satisfied only for a discrete set of eigenvalues lj with j=0,1, ..., and corresponding eigenfunctions yj(x). Additionally, the solution eigenvalues and eigenfunctions have the following properties.

• The eigenvalues lj are countably infinite and may be ordered: lj < lj+1.
• There is a smallest non-negative eigenvalue, l0 > 0. There is no greatest eigenvalue.
• Eigenfunctions, yj, posess nodes between a and b, the number of such nodes increases with increasing j. The eigenfunction y0(x) has no nodes, y1(x) has one node, and so forth.

For a complete solution of the Strum-Liouville problem, consult one the the references.

## Exercise

Show that the eigenfunctions {yj(x), j=0,1,...} are orthogonal on the interval (a,b). As part of your proof, show that the eigenvalues have to be real. It also follows that the eigenfunctions can be chosen to be normalized.

To prove the desired property, begin by writing the eigenvalue equation:

. Now multiply by yi* (we use both yj* and to denote the complex conjugate of yj) and integrate over x between a and b, yielding . Next, interchange i and j in this expression and take its complex conjugate. Subtract the resulting equation from the starting expression to obtain . When the first term is integrated by parts and the boundary conditions are used, there results . Two conclusions can be drawn:

By choosing j = i, it follows that lj = lj* or that eigenvalues of the Sturm-Liouville problem are real.

By choosing j differs from i, it follows that the integral of the product yi(x)*yj(x) over the interval is zero: the eigenfunctions are orthogonal.

## Major Example S-L Systems

 Name, notation Applications Formula or Property Legendre, Pn(x) or Pn(cosq) Laplace's equ. in spherical polar coordinates, multipole expansion, spherical harmonics, quantum rotator Rodrigues formula, Generating function, Examples, Recurrence Relations,  Differential Equation, DE in Sturm-Liouville form, Orthonormality Hermite, Hn(x) quantum oscillator Rodrigues formula, Generating function, Examples, Recurrence Relations, Differential Equation, DE in Sturm-Liouville form, Orthonormality Laguerre, Ln(r) hydrogen atom radial factor Rodrigues formula, Generating function, Examples, Recurrence Relations, Differential Equation, DE in Sturm-Liouville form, Orthonormality Bessel's function of integral order, Jn(r) Laplace's equ. in cylindrical coordinates Series Expansion, Generating function, Differential Equation, DE in Sturm-Liouville form, Orthonormality Circular functions, sin(nq), cos(nq) classical oscillator Series Expansion, Generating function, Differential Equation, DE in Sturm-Liouville form, Orthonormality

## References

• F. W. Byron & R. W. Fuller, Mathematics of Classical and Quantum Physics [Addison-Wesley, 1970]
• J. D. Jackson, Mathematics for Quantum Mechanics [W.A. Benjamin, 1962]
• M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions [Dover, 1965]

Home

Created or up-dated 08/03/99   by R.D. Poshusta