Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-10-23 DOI:10.3233/asy-231872

Jacob Shapiro

引用次数: 0

Abstract

We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity.

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在原点处具有奇点的短程L∞势的半经典解界

对于h，我们考虑E >0，半经典Schrödinger算子- h2 Δ + V−e的解析估计。接近无穷时，势的形式为V = V L + V S，其中V L是一个长范围势，它是关于径向变量的Lipschitz势，而V S = O (| x |−1 (log | x |)−ρ)对于某些ρ >1. 在原点附近，如果0≤β <，则| V |可能表现为| x |−β;2(3−1)。我们发现，对于任意ρ≈>1、有C, h 0 >对于所有h∈(0,h 0)，我们有一个形式为exp (C h−2 (log (h−1))1 + ρ≈的可解界。如果V S以更快的速度向无穷远处衰减，边界的h依赖性就会提高。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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