**Abstract:** We generalize a recent result of A. Eremenko and A. Gabrielov on
irreducibility of the spectral discriminant for the Schrdinger equation with quartic
potentials. We consider the eigenvalue problem with a complex-valued polynomial
potential of arbitrary degree d and show that the spectral determinant of this
problem is connected and irreducible. In other words, every eigenvalue can be reached
from any other by analytic continuation. We also prove connectedness of the parameter
spaces of the potentials that admit eigenfunctions satisfying k>2 boundary conditions,
except for the case d is even and k = d/2. In the latter case, connected components
of the parameter space are distinguished by the number of zeros of the eigenfunctions.
This is a joint work with Andrei Gabrielov.