Consider a simple problem $$ \frac{d^2y}{dx^2} y = -\lambda^2 y.$$ I would like to calculate the eigenvalues of operator $\, d_x^2\, $ using finite difference, but I am not sure how to do it. Discretising the problem by using $N$ internal nodes, and using the finite difference approximation
$$ \frac{d^2y}{dx^2} \Big |_{\textrm{at node } k} = \frac{y_{i+1} - 2 y_{i}+y_{i-1}}{h^2} \quad \textrm{here,} \quad h = \frac{1}{N+1}$$
One can convert the above problem to linear algebra problem $\mathbf{L}y = -\lambda^2 y $ ( Let's assume that the boundary conditions are appropriately taken into account). How are the eigenvalues of $\mathbf{L}$ are related to the eigenvalues of the operator $ \frac{d^2}{dx^2}$? Is it even a right approach to use finite difference to calculate the eigenvalues of the differential operator?
I have come across this example : How to numerically calculate a eigenvalue problem? and Solve the eigenvalue problem $y''=\lambda y$ numerically
In the first post, Where the author has raised interested question : "It seems that Finite difference method do not need to find basis. Is this a trick to solve this eigenvalue problem or still related to some kind of basis I am not aware of?" Exactly the same questions I am wondering about, but in that post, this particular question went unanswered.