Lower bounds for Dirichlet Laplacians and uncertainty principles

Stollmann, Peter; Stolz, Günter

doi:10.4171/jems/1055

Cited by 5 publications

(2 citation statements)

References 32 publications

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“…Such estimates bear various names depending on the area of analysis where they appear. For instance, quantitative unique continuation estimate, see, e.g., [RL12,LM20], uncertainty principle, see, e.g., [SS21], or spectral inequality (in the context of control theory), see, e.g., [RL12,LL21]. It is also closely related to the notion of vanishing order, see, e.g., [DF88,LL21], and annihilating pairs in Fourier analysis, see for instance [HJ94, BJPS21, ENS + 20].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density

Alexander¹,

Seelmann²,

Veselić³

2022

Preprint

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We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset S ⊂ R d ensuring that the L 2 -seminorm associated to S is equivalent to the full L 2 -norm on R d when restricted to the space of Hermite functions up to a given degree. We give precise estimates how the equivalence constant depends on this degree and on geometric parameters of S. From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is nullcontrollable from S.In all our results, the set S may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.

show abstract

Section: Introductionmentioning

confidence: 99%

Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density

Alexander¹,

Seelmann²,

Veselić³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Such estimates bear various names depending on the area of analysis where they appear. For instance, quantitative unique continuation estimate, see, e.g., [15,16], uncertainty principle, see, e.g., [23], or spectral inequality (in the context of control theory), see, e.g., [14,15]. It is also closely related to the notion of vanishing order, see, e.g., [4,14], and annihilating pairs in Fourier analysis, see for instance [1,5,9].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Principle for Hermite Functions and Null-Controllability with Sensor Sets of Decaying Density

Alexander

Seelmann

Veselić

2023

J Fourier Anal Appl

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We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset $$S\subset \mathbb {R}^d$$ S ⊂ R d ensuring that the $$L^2$$ L 2 -seminorm associated to S is equivalent to the full $$L^2$$ L 2 -norm on $$\mathbb {R}^d$$ R d when restricted to the space of Hermite functions up to a given degree. We give precise estimates how the equivalence constant depends on this degree and on geometric parameters of S. From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is null-controllable from S. In all our results, the set S may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.

show abstract

Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials

Alexander

Rose

Seelmann

et al. 2023

Journal of Differential Equations

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Lower bounds for Dirichlet Laplacians and uncertainty principles

Cited by 5 publications

References 32 publications

Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density

Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density

Uncertainty Principle for Hermite Functions and Null-Controllability with Sensor Sets of Decaying Density

Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials

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