TLDR: equation for eigenvalues of differential operator seem inverted based on what one would obtain from solving the eigenvector/-value problem. I don't understand where this equation comes from.
Hi, I'm trying to understand an analysis of a simple model PDE constrained inverse problem from [ref given at end of post]: the inference of the source term $m(x)$ of a Poisson equation with constant coefficient $k > 0$, $$ -k\frac{\partial^2 u(x) }{\partial x^2} = m(x), 0 < x < L, $$ $$ u(0) = u(L) = 0, $$ from an observation $d(x)$ of the state $u(x)$ everywhere in the domain $(0,L)$.
The authors define an parameter-to-observable map $F$ that maps the source $m(x)$ to the observable $u(x)$ as $F(m) := u(x)$ where u(x) satisfies the PDE and BCs for a given $m(x)$. They then go on to state that $F$ is a self-adjoint operator with eigenfunctions $v_j(x),j =1,2,...,\infty,$ given by: $$v_j(x) = \sqrt\frac{2}{L}sin\left(\frac{j\pi x}{L}\right)$$ with corresponding eigenvalues: $$\lambda_j=\frac{1}{k}\left(\frac{L}{j\pi}\right)^2. $$ The idea is to invert the forward map to solve for the source $m(x)$ given $d(x)$, the observation of $u(x)$, in order to illustrate important features of the solution (spectral decomposition of $F$, Fourier coefficients of the data, Picard condition, etc.)
My question regards the equation for the eigenvalues. I am used to seeing them written as $\lambda_j=\left(\frac{j\pi}{L}\right)^2$, which is what one would obtain from the eigenvalue problem of the homogeneous part of the Poisson equation. I do not understand why the authors write the eigenvalues in the way that they do. I'm also not sure why k is involved in the eigenvalues.
Essentially, there needs to be some decay of the eigenvalues for the rest of their analysis to make sense, however it seems to me that the eigenvalues actually increase with j. Any help would be much appreciated.
Thanks
[ref]: Ghattas, O., & Willcox, K. (2021). Learning physics-based models from data: Perspectives from inverse problems and model reduction. Acta Numerica, 30, 445-554. doi:10.1017/S0962492921000064