Math. Model. Nat. Phenom.
Volume 9, Number 5, 2014Spectral problems
|Page(s)||170 - 176|
|Published online||17 July 2014|
Schrödinger Operators on a Half-Line with Inverse Square Potentials
1 DICATAM, Sezione di Matematica,
Università di Brescia, Via Branze,
2 Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d’ Hères Cedex, France
Corresponding author. E-mail: email@example.com
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.
Mathematics Subject Classification: 47A10 / 35P05
Key words: inverse square potentials / dispersive estimates
© EDP Sciences, 2014
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