| 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.
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