Geodesic
Schrödinger Operators on a Half-Line with Inverse Square Potentials
H. Kovařík1 ; F. Truc2
1 DICATAM, Sezione di Matematica, Università di Brescia, Via Branze, 38 - 25123 Brescia, Italy
2 Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d’Hères Cedex, France
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 5, pp. 170-176

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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.
DOI : 10.1051/mmnp/20149511

H. Kovařík 1 ; F. Truc 2

1 DICATAM, Sezione di Matematica, Università di Brescia, Via Branze, 38 - 25123 Brescia, Italy
2 Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d’Hères Cedex, France 
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H. Kovařík; F. Truc. Schrödinger Operators on a Half-Line with Inverse Square Potentials. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 5, pp. 170-176. doi: 10.1051/mmnp/20149511

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