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Asymptotic Number of Scattering Resonances for Generic Schrödinger Operators

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Abstract

Let −Δ + V be the Schrödinger operator acting on \({L^2(\mathbb{R}^d,\mathbb{C})}\) with \({d\geq 3}\) odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. Let n V (r) denote the number of resonances of −Δ + V with modulus ≤  r. We show that if the potential V is generic in a sense of pluripotential theory then

$$n_V(r)=c_d a^dr^d+ O(r^{d-{3\over 16}+\epsilon}) \quad \mbox{as } r \to \infty$$

for any ε > 0, where c d is a dimensional constant.

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Authors and Affiliations

  1. UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu, 4 place Jussieu, 75005, Paris, France

    Tien-Cuong Dinh & Duc-Viet Vu

  2. DMA, UMR 8553, Ecole Normale Supérieure, 45 rue d’Ulm, 75005, Paris, France

    Tien-Cuong Dinh

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  1. Tien-Cuong Dinh
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  2. Duc-Viet Vu
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Correspondence to Tien-Cuong Dinh.

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Communicated by I. M. Sigal

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Dinh, TC., Vu, DV. Asymptotic Number of Scattering Resonances for Generic Schrödinger Operators. Commun. Math. Phys. 326, 185–208 (2014). https://doi.org/10.1007/s00220-013-1842-7

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