Ugh, this is trivial -- the real and complex components of the time independent Schrodinger equation are independent of each other if the potential is purely real. So the real component is a solution on its own with the desired eigenvalue.
Hmm, it doesn't look like you've used the hypothesis that we're in the ground state -- that's a little worrying as otherwise it would be true for all states. Do you use the hypothesis under the time reversal => complex conjugate part?
Also, clearly the stationary state is time independent...
I recently saw the Rayleigh Ritz variational approach used in spectral graph theory, so I was curious to look it up again in the quantum mechanics context. Anyway, there was a real sticking point quite quickly...
When we pick our trial wave function, because we want our overlap integrals...