# What is the best numerical algorithm for integrating the 1D Schrödinger equation?

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.

• See this physics.stackexchange.com question. The answer contain cut-and-pasteable code. – Igor Rivin Dec 19 '17 at 3:07
• @IgorRivin Thank you but that post seems to deal with Numerov's method. I was hoping for something a bit more advanced (without reinventing the wheel) – lcv Dec 19 '17 at 3:36