see e.g.

1-Numerical Solution of Differential Equations: Finite Difference and Finite Element Solution of the Initial, Boundary and Eigenvalue Problem in the Ma

Academic PrIsaac Fried

2-https://www.researchgate.net/topic/Eigenvalue-Problems/2

3-https://www.mathworks.com/help/pde/ug/pde.pdemodel.solvepdeeig.html

4-https://books.google.com/books?id=Yunh5w9VwTgC&pg=PA243&lpg=PA243&dq=Could+you+provide+a+reference+on+eigenvalue+problems+in+PDEs&source=bl&ots=5c9w4MTD6w&sig=ACfU3U3V7rYGC0fptzuzE4Y6fAHIsJHY5w&hl=ar&sa=X&ved=2ahUKEwjrwrPuiMvnAhWmzYUKHe3nCaQQ6AEwBXoECAoQAQ

It is a very complex problem because already for a PDE with two variables one obtains for the characteristic polynomial a polynomial with two variables and it is not easy to find the roots.

On this subject, You can consult the following reference: "T. Kaczorek, Two dimensional linear systems, Springer-Verlg, 1985"

Dear Marius Jonsson

I hope that the attached articles are useful in your research.

Best regards

The eigenvalues of an unbounded closed operator such as is highly dependent on the domain of definition of the operator. That is is the operator defined on all of R^n or is it defined on a bounded subset with a smooth boundary? In the former case the discrete spectrum might be empty (no eigenvalues) and In the latter case the eigenvalues are dependent on the boundary conditions. Just about any reference on elliptic partial differential operators will have information on the eigenvalues. In some cases - dependent on f and the domain of definition the operator will be Fredholm in which case it will have a discrete spectrum along with a finite dimensional kernel and co-kernel and hence a finite index. This is often the case for boundary value problem on bounded sets with a smooth boundary or compact manifolds.

All this assumes that "f" in your equation is real valued. Then the resulting operator is symmetric and given the correct domain of definition self adjoint. This case has been well explored and the theory of the spectrum well understood. The eigenvalues are but one component in the spectrum and often the existence of an eigenvalue is highly dependent on f in your equation.

However, your equation appears to be the time independent Schrödinger equation and there has been a wealth of work on the spectral analysis for this equation.

For example it describes the Hydrogen atom with f=a/norm(x) where a is a constant.

A place to start is

https://arxiv.org/pdf/1203.2344.pdf

https://mathematics.ceu.edu/sites/mathematics.ceu.hu/files/attachment/basicpage/27/mihailescu.pdf

https://www.amazon.com/Shmuel-Agmon/e/B001K7ORNS?ref=sr_ntt_srch_lnk_7&qid=1581603976&sr=8-7

https://www.amazon.com/Perturbation-Theory-Operators-Classics-Mathematics/dp/354058661X/ref=sr_1_1?Adv-Srch-Books-Submit.x=45&Adv-Srch-Books-Submit.y=11&keywords=Perturbation+theory&qid=1581606566&refinements=p_27%3AKato&s=books&sr=1-1&unfiltered=1

A pretty good discussion of the time independent Schrödinger equation

time independent Schrödinger equation is Hall book

https://www.amazon.com/s?k=quantum+theory+for+mathematicians&i=stripbooks&rh=p_27%3AHall&s=relevanceexprank&Adv-Srch-Books-Submit.x=35&Adv-Srch-Books-Submit.y=14&unfiltered=1&ref=sr_adv_b

of for a good intro.

https://www.reed.edu/chemistry/alan/Research/Bond/OneE/BoxProp/box.html