I am studying 2D quantum scattering models for my Bachelor's thesis. Somewhat like these:
,with Dirichlet ($\psi \mid_\Gamma = 0$) boundary conditions on the "walls" of the waveguide and the resonator. The waveguide is infinite in both directions, and admits the scattering solutions, corresponding to the continuous spectrum of the Hamiltonian.
I want to find scattering solutions, corresponding to some fixed energy $E$, corresponding to the continuous spectrum, that is, to solve the Helmholtz equation in the waveguide and the resonator. However, models of such type don't admit solutions in analytic form in general, that's why I am studying approximations of this model, for which analytic solutions exist. Of course, I'd have to verify my approximations somehow, that is, to compare my approximations to the "real" solutions.
I thought of solving numerically, however, all the numerical methods for solving boundary value problems I heard of deal with bounded domains, and in my case I've got infinite waveguide. I can't even choose some large value $L$ and consider the problem in the domain, $-L \le x \le L$, since I can't impose any reasonable boundary conditions when $x = -L$ or $x = L$, the wavefunction does not have to go to zero at the asymptotic region for scattering states.
What is more, the eigenspace for the energy $E$ is not one-dimensional in general (composed of linear combinations of solutions with "wave incident from the left" and "wave incident from the right", just like in 1D rectangular barrier scattering), so I'd like to get not just any solution, but with no "wave incident from the right" component.
So, my questions are:
- Is there a way to numerically solve differential equations (Helmholtz equation in particular) in an infinite domain?
- If 1 is the case, there a way to get not just some arbitrary solution for the energy $E$, but some particular ("wave incident from the left")?
- Are there any solvers available?