I'm wondering about some definitions of the eigenvalues and eigenfunctions of the laplacian operator and I would be really glad if you can help me on these definitions.
Let's make things simple. If we take a one dimensional case, the laplacian basically corresponds to the second order derivatives of a given function. Therefore, replicating typical operations that we can do over matrices, we would have:
$$\Delta f = \lambda f$$
However, here Jakobson defines the eigenvalue problem as:
$$\Delta f + \lambda f = 0$$
Basically changing the sign of the eigenvalue! Actually this seems reasonable since we know that the laplacian is a positive semidefinite matrix, but I don't get why we have to change the sign of the $\lambda$ in the eigenvalue problem in order to replicate this behavior.
Indeed, if we take a typical eigenfunction of the laplacian. Namely, $f(x) = e^{jnx}$, we can observe that:
$$ \Delta e^{jnx} = -n^2 e^{jnx}$$
which is negative! That actually is in contrast with what I have stated above, talking about the positivity of the laplacian's eigenvalues.
Finally, maybe a really simple question. We know that the laplacian is a self-adjoint operator for example wrt the typical inner product defined as:
$$ \langle f,g\rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \bar{g}(x) dx$$
Therefore, it should present orthogonal eigenfunctions for different eigenvalues. However, even the simple exponentials $e^{nx}$ appear to me as eigenfunctions of the laplacian which however are not orthogonal one to each other… there must be something really simple that I'm missing about this and any help will be really helpful!
Thank you very much!
Best Answer
Yes! you're missing the domain $\Omega$. The Laplacian needs to be defined in a domain $\Omega$ and with boundary conditions on the boundary of $\Omega$, $\partial \Omega$ (note that if $\partial \Omega = \emptyset$ no boundary conditions are needed), usually the boundary conditions are Dirichlet boundary conditions wich means that the eigenfunctions satisfy
$$\left\lbrace\begin{array}{cc} -\Delta u = \lambda u & \mathrm{in}\ \Omega\\ u = 0 & \mathrm{on} \ \partial \Omega \end{array}\right.$$
If you expect a discrete sequence of eigenvalues your domain must be of bounded volume.
In your case I interpret that your domain $\Omega$ is a bounded interval and then only complex exponentials satisfy the Dirichlet boundary conditions, otherwise if the Laplacian is defined in the whole real line (in this case both ccomplex and real exponentials satisfy the empty boundary conditions) then the spectrum is a continuum which I suppose is not your case.
Respect to the sign of the eigenvalues: the Laplacian is not positive (in the flat case), is negative and then $-\Delta$ is positive, with positive eigenvalues.