Semiclassical resolvent bounds in dimension two

Jacob Shapiro

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Semiclassical resolvent bounds in dimension two

Jacob Shapiro

Analysis of PDEs · Submitted 13 Apr 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We give an elementary proof of weighted resolvent bounds for semiclassical Schr\"odinger operators in dimension two.

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What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.74 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.83 · compares with prior results 0.85 · mentions code or data availability 0.02.

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We give an elementary proof of weighted resolvent bounds for semiclassical Schr\"odinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter, but near infinity it grows linearly. Our result covers the missing case from the work of Datchev.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We give an elementary proof of weighted resolvent bounds for semiclassical Schr\"odinger operators in dimension two.	contribution 1.00	0.96	0.77	asserted 1.00
s1	We require the potential function to be Lipschitz with long range decay.	limitation 0.68	0.41	0.04	asserted 0.99
s2	The resolvent norm grows exponentially in the inverse semiclassical parameter, but near infinity it grows linearly.	result 1.00	0.82	0.41	asserted 1.00
s3	Our result covers the missing case from the work of Datchev.	gap 0.61	0.68	0.09	asserted 1.00

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The paper

Original abstract

We give an elementary proof of weighted resolvent bounds for semiclassical Schr\"odinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter, but near infinity it grows linearly. Our result covers the missing case from the work of Datchev.

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5 arXiv papers cite this in 12 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 7 sentences without a confident label

Uses its method or tool 2 papers

At high energy, we use the exponential semiclassical resolvent estimate [ref] for a Lipschitz potential, proved by Datchev [ref] and the author [ref] , to show (Proposition [ref]) [equation] At low energy, we find (Proposition [ref]) [equation]

Local energy decay for Lipschitz wavespeeds · Introduction · 0.93
To prove Lemma [ref], we adapt the strategy appearing in [ref] .

Semiclassical resolvent bound for compactly supported $L^\infty$ potentials · Introduction · 1.00

Compares with it 1 paper

In dimension $d=2$ Shapiro [ref] proved [ref] replacing the above assumptions on $∂_r V$ with $∇V ∈L^∞$ and $|∇V |≤⟨r⟩^-1-δ_0$.

Semiclassical resolvent estimates for bounded potentials · Resolvent bounds · 0.81

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