Generating a spectrum for an PDE

Hi, I would like to generate the spectrum of the eigenvalues in complex space for a complex-valued Schrödinger equation:

In order to do so, I wanted to use MATLAB with some method. Is any one aware of a script that can generate the spectrum of this equation?

There is also a program for MATLAB called Chebfun, it has a functionality that can also resolve the eigenvalue landscape for a complex hermitian matrix:

http://www.chebfun.org/examples/linalg/EigLandscapes.html

How can I eventually use this for the PDE given above instead of the matrix?

Thanks!