  In the case of polynomial potentials all solutions to 1-D Schroedinger
equation are entire functions totally determined by loci of their roots and
their behaviour at infinity. In this paper a description of the first of the
two properties is given for fundamental solutions for the high complex energy
limit when the energy is quantized or not. In particular due to the fact that
the limit considered is semiclassical it is shown that loci of roots of
fundamental solutions are collected of selected Stokes lines (called
exceptional) specific for the solution considered and are distributed along
these lines in a specific way. A stable asymptotic limit of loci of zeros of
fundamental solutions on their exceptional Stokes lines has island forms and
there are infintely many of such roots islands on exceptional Stokes lines
escaping to infinity and a finite number of them on exceptional Stokes lines
which connect pairs of turning points. The results obtained for asymptotic
roots distributions of fundamental solutions in the semiclassical high
(complex) energy limit are of a general nature for polynomial potentials.
