I believe it to be impossible to find any non-trivial solution for this problem. The only solutions that I can see are with and .
Consider the following boundary value problem
where the operator is defined to be
and is the usual forward difference operator.
When I try to find a nontirivial real sequence satisfying with and or , I always fail.
I need help to find a suitable and for which admits a nontrivial solution .
Consider the following difference equation
Clearly, the set of eigenvalues of the equation is . Thus the least eigenvalue is .
Moreover one can check that
is a nontrivial solution of
But this is not a real sequence.
Oops! Sorry, it was quite late here (at night) when I responded and I didn't expand the L operator out correctly in my head.
I haven't checked your eigenvalues so I'll assume they're correct in which case I think you'll find multiplying your solution by will render it to a sequence of reals.
This is perfectly valid as your equation is homogeneous so you can scale up by whatever constant you like!
Hope that's of more help to you!
May be I should try to obtain some linear combinations of such solutions to make them real, but i dont think it is possible...