Schrödinger Operators on a Half-Line with Inverse Square Potentials

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

^⋆ Corresponding author. E-mail: hynek.kovarik@unibs.it

Abstract

We consider Schrödinger operators H_α given by equation (1.1) below. We study the asymptotic behavior of the spectral density E(H_α,λ) for λ → 0 and the L¹ → L^∞ dispersive estimates associated to the evolution operator e^{− itH_α}. In particular we prove that for positive values of α, the spectral density E(H_α,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^{− itH_α} depends on α. More precisely we show that the decay rate of e^{− itH_α} for t → ∞ can be made arbitrarily large provided we choose α large enough and consider a suitable operator norm.