I'm dealing with a one dimensional quantum mechanical scattering problem in a finite region, say $x\in\left[0,L\right]$. At first, this problem is defined as a finite-difference problem, i.e., the Hamiltonian is finite-dimensional and we can find its eigenstates by numerical methods. However, it turns out that for sufficiently large $x$, we can take a continuum limit and use some tricks to get out the analytic behavior of the eigenstates in that region.

Now, what I would like to do is take that asymptotic behavior, and use that data to feed into my more complicated finite-difference problem at small $x$ (all small $x$ calculations will be done numerically, I should say). The result should be an approximate eigenstate of the full Hamiltonian, but I'm not sure how to stitch together the continuum behavior with the more complicated details at small $x$.

Attempted Solution

So far I have tried the following: assume $\psi(x>x_{c})$ is a known function, where $x$ is some cutoff scale. Then at $x=x_{c}$, fix $\psi$ to be continuous and to have a continuous derivative. Then we can try to solve for $\psi(x<x_{c})$ by using the constraint $H\psi = E\psi$ (this is in fact possible because the Hamiltonian is tridiagonal). However, since I don't have perfect analytical control over the eigenvalues, this process is prone to error - it seems that a small deformation of $E$ can lead to large changes in the small $x$ behavior of $\psi$.

Hence I am in need of a numerically robust method that connects the complicated small $x$ features of my potential to the simple asymptotic behavior that I've been able to derive at large $x$. This seems like the kind of scattering problem which has been worked on in the past, but I'm not familiar with works that connect numerical and analytical solutions. Any suggestions for this are very welcome.



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