The semiclassical small-(ℎ/2π) limit of loci of roots of subdominant solutions for polynomial potentials

In this paper, a description of the small-(ℎ/2π) limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers. The considered potentials are those which provide us with the simple turning points only. Three types of Stokes graphs (SG's) associated with the potentials are investigated - the general non-critical ones, the general critical ones but with only single internal Stokes line (SL), and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized (ℎ/2π). In particular due to the fact that the small-(ℎ/2π) limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines.