Sturm Liouville Theory is significant because it allows us to solve a wide range of physical problems mathematically. This theory provides a method for solving certain types of differential equations, which are essential in many areas of physics. It is particularly relevant in quantum mechanics, where it is often used to find solutions to the Schrödinger equation. The theory also shows that the eigenfunctions of these differential operators can form an orthogonal basis, which is a very useful property for solving physical problems. It also allows for the decomposition of a second order operator in a way that is Hermitian, which is crucial in quantum mechanics.
When we first considered the eigenproblem, we only considered it for finite dimensional vector spaces. However, we can consider these same problem with continuous vector spaces. In these spaces, we can represent our linear operators as a product of functions and derivatives called linear differential operators
<aside> 💡 Linear Differential Operators
Linear Differential operators of order $N$ are of the form
$$ \hat L=\alpha_N(x)\frac{d^N}{dx^N}+\alpha_{N-1}(x)\frac{d^{N-1}}{dx^{N-1}}+\dots+\alpha_{1}(x)\frac{d}{dx}+\alpha_0(x) $$
Since they are linear they has the following property:
Suppose $f(x)$ and $g(x)$ are functions and $a_1$ and $a_2$ are scalars:
$$ \hat L[a_1f+a_2g]=a_1\hat L[f]+a_2\hat L[g] $$
With linear operators we typically drop the $[]$ implying the operator acts on the function to its right:
$$ a_1\hat Lf+a_2\hat Lg $$
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We still propose the eigenvalue problem in the same way:
$$ \hat Lf=\lambda f $$
where $\lambda$ is some scalar and $f$ is called an eigenfunction of our operator $\hat L$
We are already familiar with some eigenfunctions for differential operators. What function when we apply a derivative returns us the same function? The exponential function! This even applies to the integral operator as well. This is why exponentials make such a good guess for any constant coefficient homogenous ODE; they are the building blocks for pure constant coefficient derivative differential operators.
In these notes, we’ll focus specifically on operators that are Hermitian. Recall that operators are Hermitian under an inner product,$\big(\braket{\cdot,\cdot}\big)$ ,when
$$ \braket{g|\hat L|f}=\braket{f|\hat L^\dag|g}^=\braket{f|\hat L|g}^ $$
If this is true then it implies $\hat L=\hat L^\dag$
We will use this definition to test whether $\hat L$ is hermitian for second order linear differential operators and take a quick look on whether we can apply this to first order differential operators. Ultimately these topics fall under Functional Analysis and the Study of Differential Equations and we will only be covering the important take aways from the Sturm Liouville Theory.
These operators can be made to be Hermitian and thus the eigenfunctions for the operator form an orthonormal basis. We find the eigenfunctions by solving the eigenproblem. The eigenfunction solutions will be guaranteed to be orthogonal.
Lets say we have the operator $\hat L$ and we’ve found the weighting function $w(x)$ and applied the appropriate boundary conditions such that $\hat L$ is Hermitian. Then if we solve the following homogenous equation then the solutions $f$ will be the eigenfunctions for our operator.
$$ \hat Lf=\lambda f\\ (\hat L-\lambda)f=0\\ p(x)\frac{d^2f}{dx^2}+q(x)\frac{df}{dx}+(r(x)-\lambda)f=0 $$
Let’s define an inner product for our space of functions:
$$ \braket{f|g}=\int_I f^*(x)g(x)w(x)dx $$
By the condition for Hermitian we have: