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

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  • Jamel Ben Amara
  • Jihed Hedhly

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.

Suggested Citation

  • Jamel Ben Amara & Jihed Hedhly, 2018. "Upper bound for the ratios of eigenvalues of Schrödinger operators with nonnegative single‐barrier potentials," Mathematische Nachrichten, Wiley Blackwell, vol. 291(13), pages 1926-1940, September.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:13:p:1926-1940
    DOI: 10.1002/mana.201700164
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