The purpose of this note is to recall the theory of the (homogenized) spectral problem for Schrödinger equation with a polynomial potential and its relation with quadratic differentials. We derive from results of this theory that the accumulation rays of the eigenvalues of the latter problem are in $$1-1$$ 1 - 1 -correspondence with the short geodesics of the singular planar metrics induced by the corresponding quadratic differential. We prove that for a polynomial potential of degree $$d,$$ d , the number of such accumulation rays can be any positive integer between $$(d-1)$$ ( d - 1 ) and $$d \atopwithdelims ()2$$ d 2 .
Analysis and Mathematical Physics – Springer Journals
Published: Feb 3, 2015