Sturm-Liouville boundary conditions

The solution of a linear matrix differential equation with prescribed initial conditions is a relatively straightforward process, although the result can be complicated by the noncommutativity of the matrix of coefficients for different values of the independent variable. The solutions generally behave as exponentials, whose law of exponents is further distorted by noncommutativity. Nevertheless, the distinction between real and imaginary exponents holds, so that some solutions behave like hyperbolic functions and others behave like trigonometric functions. Those of trigonometric type oscillate, have zeroes, and remain bounded when the independent variable is real. Of course, when the solutions are taken as functions of a complex variable, all this structure rotates into the complex plane and may have to contend with branch points and other singularities.

A variant on the problem of initial values is the problem of final values. Taken over a finite interval, the nonsingularity of the solution matrix permits its inversion with an immediate solution to the problem. Asymptotically the situation is complicated by the fact that although the solution matrix is technically invertible, the difference between growing and diminishing eigenvalues is so great that only certain kinds of limiting information can be obtained.

Straddling these two extremes, the so-called Sturm-Liouville boundary conditions consist in a partial specification of initial values together with additional final values, sufficient to get as unique a solution as possible. Taken at face value, such a mixture is not likely to have a solution, but in practice, the differential equation often depends upon a parameter, usually the consequence of separating variables to solve some partial differential equation. The constant of separation would then be adjusted until the split boundary conditions could be satisfied.

That such a procedure has hope of success is encouraged when it is observed that varying the separation parameter, for given initial conditions, runs the terminal conditions through a range of values, one of which could be just the one sought after. In fact it is separation of variables which typically leads to an eigenvalue equation, because in the separated form it is the quotient of a differential operator depending on a single variable applied to its function by that separated function itself which is equal to a constant. So algebraic rearrangement sets the operator up as producing a multiple when it acts on its function:

$${\cal H}\Psi E \Psi,$$ = (340)

anticipating the notation of quantum mechanics. If ${\cal H}$ is a second order differential operator, it can be replaced by a matrix combination of a pair of first order operators.