We consider two- and three-dimensional complex Schrödinger equations with Abelian potentials and a fixed energy level. The potential, wave function, and the spectral Bloch variety are calculated in terms of the Kleinian hyperelliptic functions associated with a genus two hyperelliptic curve. In the special case in two dimensions when the curve covers two elliptic curves, exactly solvable Schrödinger equations are constructed in terms of the elliptic functions of these curves. The solutions obtained are illustrated by a number of plots. © 2002 American Institute of Physics.