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I am currently developing a numerical solver, which solves the standard, one-dimensional, time-independent Schrödinger equation

$$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi \tag{1}$$ I want to make it such that my solver works for non-uniform grids. More precisely, I would like to be able to have one grid spacing $a_{1}$ for say $x<0$ and another $a_{2}$ for $x\geq 0$. To do this I employ the following finite difference discretization of (1)

$$\frac{-\hbar^2}{2m} \Bigl( \frac{\psi_{i+1}-\psi_{i}}{x_{i+1}-x_{i}} - \frac{\psi_{i}-\psi_{i-1}}{x_{i}-x_{i-1}} \Bigr)\cdot \frac{2}{x_{i+1}-x_{i-1}} + V_{i}\psi_{i} = E\psi_{i}\tag{2}$$

This can then be solved as an eigenvalue equation $\hat{H}\boldsymbol{\psi} = E\boldsymbol{\psi}$, and it turns out that this delivers consistent results, also for non-uniform grid spacings.

There is however one thing that makes me uncertain about the validity of (2). Namely that when applying it to a non-uniform grid the resulting Hamiltonian matrix $\hat{H}$ is not hermitian, which is evident if one considers a point lying on the boundary between grid points with spacing $a_{1}$ and grid points with spacing $a_{2}$. In this case $\hat{H}$ will have a entries, which do not match their corresponding hermitian conjugate entries.

Is this a problem? And if so, what would be the correct way to discretize (1) and preserve hermiticity?

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Ok, this answer comes a bit late, but I just bumped into the exact same question and found a elegant solution in a paper by Tan, Snider et al available here:

https://pdfs.semanticscholar.org/6128/2f07c3a3cf778355582943b0412891add20c.pdf

Ref: Tan, Snider, Chuang & Hu, J. Appl. Phys. 68, 4071 (1990).

They transform Eq.(2) above into an hermitian matrix by multiplying and dividing it by $L_i = \sqrt{x_{i+1}-x_{i-1}}$.

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