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inverse square potentials

Mathematical Modelling of Natural Phenomena (1)

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Schrödinger Operators on a Half-Line with Inverse SquarePotentials
H. Kovařík, F. Truc
Journal:

Mathematical Modelling of Natural Phenomena / Volume 9 / Issue 5 / 2014

Published online by Cambridge University Press:

17 July 2014, pp. 170-176

Print publication:

2014
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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 andthe L1 →L∞ dispersive estimates associated to theevolution operator e−itHα.In particular we prove that for positive values of α, the spectral densityE(Hα,λ)tends to zero as λ →0 with higher speed compared to the spectral density of Schrödingeroperators with a short-range potential V. We then show how the long time behavior ofe−itHαdepends on α.More precisely we show that the decay rate of e−itHαfor t → ∞ canbe made arbitrarily large provided we choose α large enough and consider a suitable operatornorm.