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

TL;DR

This paper revisits the spectral theory of Schrödinger equations with polynomial potentials, highlighting the connection between eigenvalue accumulation rays and quadratic differentials, and characterizing their possible counts.

Contribution

It clarifies the relationship between eigenvalue accumulation rays and quadratic differentials, extending the understanding of spectral problems with polynomial potentials.

Findings

Accumulation rays correspond to short geodesics of quadratic differentials.

Number of accumulation rays ranges from (d-1) to d choose 2 for degree d potentials.

The paper links spectral theory with geometric structures on the Riemann sphere.

Abstract

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.