In my "Numerical methods in Linear Algebra" course I have to calculate the eigenvalues and eigenfunctions/eigenvectors for the 1-D stationary Schrodinger equation. The interval where the $x$ axis is discretised is $[-a, a]$. As the number of gridpoints I have $N$, so the distance between two points $x_i$ and $x_{i+1}$ is $\Delta x = \frac{2a}{N}$. One of the tasks is to extrapolate the eigenfunctions to $\Delta x \to 0$ and $a \to \infty$.
I know that the difference between interpolation and extrapolation is that in the case of interpolation, the desired $x$ is in between the some given points $x_0$ and $x_{N-1}$, where we have $x_0 \lt x_1 \lt .... \lt x_{N-1}$ and extrapolation would be then the desired $x$ is outside the region of $x_0$ and $x_{N-1}$.
But I'm stuck with the extrapolation to infinity. Could somebody explain how to do this?
One example of an eigenfunction/eigenvector that i have is $[7.15552891e-09, \ 4.08699622e-06,\ 1.27412046e-03, \ 9.49478867e-02, \ 7.00702237e-01, \ 7.00701752e-01, \ 9.49477897e-02, \ 1.27409120e-03]$