Daniel,

1.) You can use complex valued algebra if you split your problem into two 
equations. Apart from that most of the algebra objects can be used with 
std::complex<double>.
At the moment, this is not true for constraints given by a ConstraintMatrix 
object, but this is WIP (https://github.com/dealii/dealii/issues/1760).
2.) Using SLEPc it should be able to compute eigenvalues for a complex 
valued system
3.) As already mentioned at the moment you can't use std::complex in a 
ConstraintMatrix. If however, you are able to formulate your problem as a 
system of real valued problems 
you can use whatever (linear) boundary condition you want. You can use 
make_periodicity_constraints to setup the DoFs you want to constraint and 
modify the inhomogeneity afterwards suitably.
4.) If you can formulate your problem as PDE (in not more than 3D) you can 
likely use deal.II for solving it numerically.

Best,
Daniel

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