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

The high energy semiclassical asymptotics of loci of roots of fundamental solutions for polynomial potentials