Suppose we have a differential operator like a quantum mechanical Hamiltonian:
$$\hat H=-\nabla^2+U$$ with zero Dirichlet boundary conditions.
In one dimension its eigenvalues can be easily found using e.g. shooting method, storing only a few numbers at any stage of computation. Once the eigenvalue is found to desired accuracy, it's trivial to sample the eigenfunction on some (possibly sparse) grid to high precision.
In $N>1$ dimensions this approach wouldn't work, because solving a Cauchy problem requires at least storing $N-1$-dimensional function on each stage.
So, my question is: is there any method, which would make it in principle possible to compute at least eigenvalues, better if also eigenfunctions of $N$-dimensional differential operator to arbitrary precision, without requiring to store any more than a constant number of values (possibly depending on $N$, but not on precision desired)?
If not in general, then maybe there's something like that for some specific $N>1$, like $N=2$?
The difference between $N=1$ and $N>1$ is that the equation is an ODE for $N=1$ whereas it is a PDE for $N>1$. So it is not surprising that for $N=1$ there are much more efficient methods. I am not aware of a way to numerically solve a PDE without storing the function on a grid. And it seems unlikely that one can solve the eigenvalue problem without solving first the PDE.