It seems like, when studying functions defined on some space (e.g. $\mathbb{R} / 2\pi\mathbb{Z}$, $S^2$) and looking for a basis of the space of such functions, the eigenvectors of the Laplacian are often chosen.
For instance:
- The Fourier series is a decomposition of functions on $\mathbb{R} / 2\pi\mathbb{Z}$ on a basis of eigenvectors of the Laplacian
- The same is done on $S^2$ using the spherical harmonics (which are eigenvectors of the Laplacian on the sphere).
My question is: what are the reasons for using the eigenvectors of the Laplacian when decomposing such functions?
I can see how it can be useful when studying solutions to equations involving the Laplacian (e.g. heat equation, wave equation), but it seems the usage of those basis goes beyond this context. What are the other reasons for using them? (compared to other orthogonal basis)
I think this can be best understood from a physics perspective.
In quantum mechanics, wave functions $\psi$ satisfy Schrodinger's time-independent equation $$\frac{-\hbar^2}{2m} \nabla^2\psi\ +V\psi=E\ \psi$$ where $V$ is a scalar function that induces boundary conditions (BC) on our allowed solutions. Setting aside our BCs, if $V$ is small compared to our energy $E$, then we can also consider $V$ as a perturbation, and approximate the allowed energies of our equation to be the same as if the $V=0$ (but not dropping our BCs). This is a $0$th order approximation; we can find higher order corrections using a Taylor series. So, we see that in many instances the solutions can be approximated to be the eigenfunctions of the Laplace operator. Additionally, the use of variational calculus on Hermitian operators (e.g. the Laplace operator + $V$) allows us to use test functions to obtain over-approximations; often, the eigenfunctions of the Laplace operator are fairly accurate test functions.
The solutions of the Laplace operator can also have physical meaning; specifically, the sine functions $\sin(n\pi x/L)$ represent oscillations of discrete frequencies. Thus, if we have a music sample, then the Fourier components are more than abstract linear algebra; they represent the "amount" of music with the notes C,C#, D..., for all the notes in music theory. If we are studying data, and want to extract the periodicity of certain phenomena, than a continuum of functions are used; this is the Fourier transform.