Is the Sturm-Liouville Operator Symmetric in Inner Product Spaces?

[Poirot1]

What does it mean for Sturm-Liouville operator to be symmetric w.r.t an inner product?

I was reading in a book that it is symmetric but that was about a certain integral being zero and inner products had not even been mentioned.

 

[Ackbach]

You have to generalize the concept of "inner product". In normal 3D-space, it's called the dot product:
$$\langle \vec{x}|\vec{y}\rangle=\vec{x}\cdot\vec{y}=\sum_{j=1}^{3}x_{j}y_{j}.$$
But there are other "spaces" out there: metric spaces, normed spaces, inner product spaces, Banach spaces, Hilbert spaces, Sobolev spaces. They each have different axioms with which you start. The inner product space is fairly general, and the "vectors" can be the usual vectors in 3D space, or they could be functions in a function space. The usual inner product defined in a function space is
$$\langle f|g\rangle:=\int_{A} \overline{f} \,g\,d\mu.$$
Here $\overline{f}$ indicates the complex conjugate of $f$, and the $d\mu$ indicates that we've defined this integral to be a Lebesgue integral. $A$ is the set over which the function space is defined.

Now we have the background to answer your question. A Sturm-Liouville operator
$$L=\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\,\frac{d}{dx}\right]+q(x)\right)$$
is symmetric (more properly, Hermitian) w.r.t. the inner product
$$\langle f|g\rangle:=\int_{A}\overline{f}\,g\,w(x)\,d\mu,$$
if and only if for every $f, g$ in the inner product space, it is true that
$$\langle Lf|g\rangle=\langle f|Lg\rangle.$$
With the Sturm-Liouville operator, you need to show that
$$\int_{A}\overline{\left\{\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\,\frac{df}{dx}\right]+q(x)f\right)\right\}}\,g\,w(x)\,d\mu=\int_{A} \overline{f} \left\{\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\,\frac{dg}{dx}\right]+q(x)g\right)\right\}\,w(x)\,d\mu.$$

You can do this using simple integration by parts twice. The boundary terms vanish because of the conditions on them. (Note that the boundary conditions are considered to be part of the operator.)

I note you marked this thread as solved. That is good. Perhaps this post will throw in a few helpful concepts.

 

[Poirot1]

Sorry you took your time. I should perhaps do a bit more reasearch before asking here.

 

[Ackbach]

Sorry you took your time. I should perhaps do a bit more reasearch before asking here.

No, that's all right. Incidentally, there are some concepts here that might help you with your other problem. Take a look at how $w(x)$ appears in the Sturm-Liouville operator, as well as how it shows up in the inner product w.r.t. which the Sturm-Liouville operator is symmetric.

 

[blue_raver22]

The Sturm-Liouville operator is a mathematical operator that is commonly used in the study of differential equations and boundary value problems. It is defined as a second-order differential operator that involves a variable coefficient and a weight function.

When we say that the Sturm-Liouville operator is symmetric with respect to an inner product, it means that the operator satisfies a specific symmetry property when applied to functions in the inner product space. This property is known as self-adjointness and is closely related to the concept of symmetry.

In simpler terms, if we have two functions f(x) and g(x) in the inner product space, the Sturm-Liouville operator being symmetric means that the inner product of the operator applied to f(x) and g(x) is equal to the inner product of g(x) and the operator applied to f(x). This is similar to the concept of mirror symmetry, where the reflection of an object is identical to the original.

The significance of this property is that it allows us to use powerful mathematical tools, such as the spectral theorem, to study the properties of the Sturm-Liouville operator and its associated differential equations. It also allows us to make important conclusions about the eigenvalues and eigenfunctions of the operator, which have important applications in physics and engineering.

In summary, the symmetry of the Sturm-Liouville operator with respect to an inner product is a fundamental property that plays a crucial role in the analysis of differential equations and boundary value problems.