Some Quantitative Unique Continuation Results for Eigenfunctions of the Magnetic Schrödinger Operator

Davey, Blair

doi:10.1080/03605302.2013.796380

Cited by 43 publications

(55 citation statements)

References 15 publications

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“…The proof in [BK05] is based on the Carleman method. In the spirit of the Carleman method, several extensions have been made in [CS99,Dav14,DZ17,DZ18,LW14], which also take singular drift coefficients and potentials into account.…”

Section: Introductionmentioning

confidence: 99%

On the fractional Landis conjecture

Rüland

Wang

2019

Journal of Functional Analysis

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In this paper we study a Landis-type conjecture for fractional Schrödinger equations of fractional power s ∈ (0, 1) with potentials. We discuss both the cases of differentiable and non-differentiable potentials. On the one hand, it turns out for differentiable potentials with some a priori bounds, if a solution decays at a rate e −|x| 1+ , then this solution is trivial. On the other hand, for s ∈ (1/4, 1) and merely bounded non-differentiable potentials, if a solution decays at a rate e −|x| α with α > 4s/(4s − 1), then this solution must again be trivial. Remark that when s → 1, 4s/(4s − 1) → 4/3 which is the optimal exponent for the standard Laplacian. For the case of non-differential potentials and s ∈ (1/4, 1), we also derive a quantitative estimate mimicking the classical result by Bourgain and Kenig.Wang is supported in part by MOST

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Section: Introductionmentioning

confidence: 99%

On the fractional Landis conjecture

Rüland

Wang

2019

Journal of Functional Analysis

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“…There are many works concerning unique continuation for Schrödinger operators with magnetic fields in the case of one-particle systems, based on Carleman estimates [2,8,24,42,43,57,58]. Another way of proving strong UCP results relies on techniques developed by Garofalo and Lin [14,15] which do not employ Carleman estimates but Almgren's monotonicity formula [1].…”

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confidence: 99%

Unique Continuation for Many-Body Schrödinger Operators and the Hohenberg-Kohn Theorem

Garrigue

2018

Math Phys Anal Geom

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We prove the strong unique continuation property for manybody Pauli operators with external potentials, interaction potentials and magnetic fields in L p loc (R d ), and with magnetic potentials in L q loc (R d ), where p > max(2d/3, 2) and q > 2d. For this purpose, we prove a singular Carleman estimate involving fractional Laplacian operators. Consequently, we obtain the Hohenberg-Kohn theorem for the Maxwell-Schrödinger model.

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“…Moreover, Kenig in [Ken07] pointed out that the exponent 2 3 of K 2 3 0 is optimal for complex valued potential function V (x) based on Meshkov's example in [Mes92]. Especially, if the real valued potential function V (x) ≥ 0, Kenig, Silvestre and Wang [KSW15] were able to show that the vanishing order is less than CK with W (x) L ∞ ≤ K 1 , Bakri in [Bak13] and Davey in [Dav14] independently generalized the quantitative uniqueness result and obtained that the order of vanishing is less than C(K 2/3 0 +K 2 1 ). The strong unique continuation property also holds for second order elliptic equation (1.5) with singular lower terms in L r Lebesgue space, i.e.…”

Section: Introductionmentioning

confidence: 99%

Quantitative uniqueness of solutions to parabolic equations

Zhu

2018

Journal of Functional Analysis

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We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize quantitative uniqueness by the rate of vanishing. We can obtain the vanishing order of solutions by C 1,1 norm of the potential functions, as well as the L ∞ norm of the coefficient functions. Some quantitative Carleman estimates and three cylinder inequalities are established. 1 2 independently by Bakri in [Bak12] and Zhu in [Zhu16]. It matches the optimal result for the vanishing order of eigenfunctions in Donnelly and Fefferman's work in [DF88].

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Some Quantitative Unique Continuation Results for Eigenfunctions of the Magnetic Schrödinger Operator

Cited by 43 publications

References 15 publications

On the fractional Landis conjecture

On the fractional Landis conjecture

Unique Continuation for Many-Body Schrödinger Operators and the Hohenberg-Kohn Theorem

Quantitative uniqueness of solutions to parabolic equations

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