Study Sturm-Liouville Eigenvalue Problems

[jwsiii]

I am studying Sturm-Liouville eigenvalue problems and their eigenfunctions form a "complete set". Can someone explain to me what this means?

 

[HallsofIvy]

A "complete" set of eigenvectors is a basis for the vector space consisting entirely of eigenvectors for a given linear transformation. If, for example, in a finite dimensional vector space, you can find a complete set of eigenvectors for a linear transformation, using those eigenvectors as the basis, you can write the linear transformation as a diagonal matrix, simplifying the problem greatly. It can be shown that "self-adjoint" linear operators always have a complete set of eigenvectors.

In working with linear differential equations, the underlying vector space is the space of infinitely differentiable functions which is infinite dimensional but using the "eigenfunctions" as a basis will still simplify the problem. The Sturm-Liouville differential operators are precisely the self-adjoint operators in that space. The simplest example is the differential operator $$\frac{d^2}{dx^2}$$ with x between 0 and $$\pi$$. It is easy to show that the eigenfunctions are cos(nx), sin(nx) and using those as a basis gives the Fourier series for a function.

 

[M98Ranger]

Sure, I'd be happy to explain what it means for eigenfunctions to form a "complete set" in the context of Sturm-Liouville eigenvalue problems.

First, let's define what we mean by eigenfunctions. In a Sturm-Liouville eigenvalue problem, we are looking for functions that satisfy a certain differential equation, along with certain boundary conditions. These functions are called eigenfunctions, and they have the property that when they are multiplied by a constant (called an eigenvalue), the resulting function is equal to the derivative of the original function.

Now, when we say that eigenfunctions form a "complete set", we mean that any function that satisfies the same differential equation and boundary conditions can be expressed as a linear combination of these eigenfunctions. In other words, the eigenfunctions form a basis for the space of solutions to the Sturm-Liouville problem.

This is a very powerful property, as it allows us to express any function that satisfies the same problem in terms of these eigenfunctions. This can greatly simplify the analysis of the problem and make it easier to find solutions.

In addition, the completeness of the eigenfunctions also means that we can use them to approximate any function that satisfies the same problem. This is because we can choose a finite number of eigenfunctions and their corresponding eigenvalues to construct an approximation of the original function.

Overall, the fact that eigenfunctions form a complete set in Sturm-Liouville eigenvalue problems is a fundamental and useful property that allows us to understand and solve these problems in a more efficient manner.