Real Analyticity of Solutions to Schrödinger Equations Involving a Fractional Laplacian and Other Fourier Multipliers

Dall’Acqua, Anna; Fournais, Søren; Sørensen, Thomas Østergaard; Stockmeyer, Edgardo

doi:10.1142/9789814449243_0062

Cited by 4 publications

(6 citation statements)

References 3 publications

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“…(6) Regularity of eigenfunctions. Despite the non-locality of the fractional Laplacian, eigenfunctions of (−∆) s + V can be shown to be regular where V is regular [34,35]. For improved Hölder continuity results for radial potentials, see [78].…”

Section: Some Further Topicsmentioning

confidence: 99%

Eigenvalue Bounds for the Fractional Laplacian: A Review

Frank¹

2017

Recent Developments in Nonlocal Theory

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Section: Some Further Topicsmentioning

confidence: 99%

Eigenvalue Bounds for the Fractional Laplacian: A Review

Frank¹

2017

Recent Developments in Nonlocal Theory

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“…This follows directly from the orthogonality of the decomposition ( 7). (iii) The regularity assumption (10) replaces the assumption in Theorem 3.1 that V ψ, f ∈ L N/(2s) (R N ), see also Remark 3.2 (ii). (The function spaces appearing in it are the natural generalizations of the ones from Definition 2.2 to more general measure spaces.)…”

Section: Improved Regularity For Radial V and Fmentioning

confidence: 99%

“…We were motivated in part by the papers [10,11] where analyticity of atomic eigenfunctions away from the nuclei is shown in the pseudorelativistic regime. In terms of methods, we draw on a helpful localization trick due to Silvestre [37] which was further developed by Felmer, Quaas and Tan [14].…”

Section: Introductionmentioning

confidence: 99%

On the Hölder regularity for the fractional Schrödinger equation and its improvement for radial data

Lemm

2016

Communications in Partial Differential Equations

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We consider the linear, time-independent fractional Schrödinger equationWe are interested in the local Hölder exponents of distributional solutions ψ, assuming local L p integrability of the functions V and f . By standard arguments, we obtain the formula 2s − N/p for the local Hölder exponent of ψ where we take some extra care regarding endpoint cases. For our main result, we assume that V and f (but not necessarily ψ) are radial functions, a situation which is commonplace in applications. We find that the regularity theory "becomes one-dimensional" in the sense that the Hölder exponent improves from 2s − N/p to 2s − 1/p away from the origin. Similar results hold for ∇ψ as well.

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“…Further analyticity and Gevrey regularity results in the context of non-local differential equations can be found for example in [DFSS12,DFSS14,AFV15,Bla20a,Bla20b].…”

Section: Introductionmentioning

confidence: 96%

On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case

Steenebrügge,

Vorderobermeier

2021

Preprint

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We prove the analyticity of smooth critical points for generalized integral Menger curvature energies intM (p,2) , with p ∈ ( 7 3 , 8 3 ), subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical C 1 -curves γ : R/Z → R n of generalized integral Menger curvature intM (p,2) subject to a fixed length constraint are not only C ∞ but also analytic. Our approach is inspired by analyticity results on critical points for O'Hara's knot energies based on Cauchy's method of majorants and a decomposition of the first variation. The main new idea is an additional iteration in the recursive estimate of the derivatives to obtain a sufficient difference in the order of regularity.

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Real Analyticity of Solutions to Schrödinger Equations Involving a Fractional Laplacian and Other Fourier Multipliers

Abstract: We prove analyticity of solutions in R n , n ≥ 1, to certain nonlocal linear Schrödinger equations with analytic potentials.

Cited by 4 publications

References 3 publications

Eigenvalue Bounds for the Fractional Laplacian: A Review

Eigenvalue Bounds for the Fractional Laplacian: A Review

On the Hölder regularity for the fractional Schrödinger equation and its improvement for radial data

On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case

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