MATSLISE: (Regular) Sturm-Liouville equation    

The File-Menu

New Problem

Closes the current Sturm-Liouville problem and opens a new problem: a Schrödinger problem, a new Sturm-Liouville problem or a a distorted Coulomb problem.

Open Problem

Displays a dialog box that enables you to retrieve a problem which was saved earlier.

Some (test)problems are predefined in the directory predefined_problems.

Save Problem

Allows you to save the problem you defined. This saved problem can then be re-opened later. All problems should be saved in an .mat-file.

Please note that information about the problem can be entered in the text-field at the bottom of the Sturm-Liouville window. This information is also saved with the rest of the problem.


The Options-Menu

Show partition

When this option is checked, an additional plot is produced by the construct-button. On this plot the partition is displayed for the associated Schrödinger problem.

Half-range reduction

Half-range reduction can be applied to symmetric Sturm-Liouville problems. A Sturm-Liouville problem is symmetric when the problem is posed on the interval -b to b, where b may be inf, and the coefficient functions are even,

p(x)=p(-x),    q(x)=q(-x),    w(x)=w(-x),

and the boundary conditions are similarly symmetric, which means that

A0 = A1,     B0 = -B1.

In this case the eigenfunctions belonging to eigenvalue E_k (k=0,1,...) are even or odd functions according as k is even or odd. Hence, the eigenvalues can be obtained by solving the given equation, but on the interval [0,b], with the given boundary condition at b and with

y'(0) = 0 to get the even eigenvalues
y(0) = 0 to get the odd eigenvalues.

The normalized eigenfunctions of the full-range problem are reconstructed from those of the half-range problem by extending in the appropriate way and dividing by sqrt(2).

Reference: J. D. Pryce, Numerical Solution of Sturm-Liouville Problems (Oxford Univ. Press, Oxford, 1993), §2.5.2.

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