A Note on EVgrafov-Fedoryuk's theory and quadratic differentials

The purpose of this short paper is to recall the theory of the (homogenized) spectral problem for a Schroedinger equation with a polynomial potential developed in the 60's by M. Evgrafov with M. Fedoryuk, and, by Y. Sibuya and its relation with quadratic differentials. We derive from these results that the accumulation rays of the eigenvalues of this problem are in 1-1 -correspondence with the short geodesics of the singular planar metrics on CP^1 induced by the corresponding quadratic differential. Using this interpretation we show that for a polynomial potential of degree d the number of such accumulation rays can be any positive integer between (d-1) and d \choose 2.