I'm interested in numerical algorithms for 1-dimensional Hamiltonians of the form

$$ H = -\frac{d^2}{dx^2} + V(x) \quad \quad (1) $$

defined on the line ($x\in\mathbb{R}$) or on the circle. The potential $V$ is such that the lowest part of the spectrum is made of eigenvalues.

My definition of *best* (and of 'integrating') is that the algorithm should return more accurately the lowest eigenvalues and corresponding eigenfunctions (say the lowest $N$ with $N$ of order 10-20).

I'm aware of *Numerov's method* but I think there should be better algorithms given my desiderata.

In particular I wonder wether Krylov methods exist for my problem. The potential $V$ is given explicitly so computing $H \psi$ for a nice smooth $\psi$ is easy and in general obtaining $\langle \phi, H^n \psi \rangle$ seems possible to a certain degree of accuracy.

Bonus points if the algorithm is implemented in some (available) software package.