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Schrödinger Operators on a Half-Line with Inverse Square Potentials

Published online by Cambridge University Press:  17 July 2014

H. Kovařík  and
F. Truc
H. Kovařík*
Affiliation:
DICATAM, Sezione di Matematica, Università di Brescia, Via Branze, 38 - 25123 Brescia, Italy
F. Truc
Affiliation:
Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d’Hères Cedex, France
*
⋆ Corresponding author. E-mail: hynek.kovarik@unibs.it
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Abstract

We consider Schrödinger operators Hα given by equation (1.1) below. We study the asymptotic behavior of the spectral density E(Hα,λ) for λ → 0 and the L1 → L∞ dispersive estimates associated to the evolution operator e− itHα. In particular we prove that for positive values of α, the spectral density E(Hα,λ) tends to zero as λ → 0 with higher speed compared to the spectral density of Schrödinger operators with a short-range potential V. We then show how the long time behavior of e− itHα depends on α. More precisely we show that the decay rate of e− itHα for t → ∞ can be made arbitrarily large provided we choose α large enough and consider a suitable operator norm.

Keywords

inverse square potentialsdispersive estimates
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 170 - 176
Copyright
© EDP Sciences, 2014

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