Upper bound for the ratios of eigenvalues of Schrödinger operators with nonnegative single‐barrier potentials

Jamel Ben Amara and Jihed Hedhly

Mathematische Nachrichten, 2018, vol. 291, issue 13, 1926-1940

Abstract: In this paper we prove the optimal upper bound λnλm≤n2m2λn>λm≥11supx∈[0,1]q(x)for one‐dimensional Schrödinger operators with a nonnegative differentiable and single‐barrier potential q(x), such that ∣q′(x)∣≤q∗, where q∗=215min{q(0),q(1)}. In particular, if q(x) satisfies the additional condition supx∈[0,1]q(x)≤π211, then λnλm≤n2m2 for n>m≥1. For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.

Date: 2018
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