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. 



What is the best numerical algorithm for integrating the 1D Schrödinger equation?
To obtain the lowest eigenvalues by means of a [Krylov subspace](https://en.wikipedia.org/wiki/Krylov_subspace) method you could use the [Lanczos algorithm,](https://en.wikipedia.org/wiki/Lanczos_algorithm) as explained for example in:


[Solving the discretized time‐independent Schrödinger equation with the Lanczos procedure](http://aip.scitation.org/doi/abs/10.1063/1.458575)



> 
> A new method is presented to find bound state solutions of the one‐,
>  two‐, or three‐dimensional Schrödinger equation. The equation is
>  turned into a sparse matrix eigenvalue problem by representing the
>  potential energy surface and the wave function on a grid. The
>  Laplacian is represented by a high (10th) order finite difference
>  formula. Eigenvalues are found by the Lanczos procedure. Examples are
>  given for the 1D Morse oscillator and the 2D Hénon‐Heiles potential.
>  Numerical convergence can be checked easily and highly accurate
>  results can be obtained. The algorithm is fast, easy to implement, and
>  vectorizable.
> 
> 
> 


