The sharp Hardy uncertainty principle for Schrödinger evolutions

Escauriaza, Luis; Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis

doi:10.1215/00127094-2010-053

Cited by 68 publications

(146 citation statements)

References 13 publications

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“…Mathematical investigations of generalized uncertainty relations can be found, for example, in [11,12]. In this subsection, we derive a generalized uncertainty relation for an arbitrarily shaped quantum dot in d dimensions, by considering the nonnegative integral…”

Section: Generalized Uncertainty Relationmentioning

confidence: 99%

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From a particle in a box to the uncertainty relation in a quantum dot and to reflecting walls for relativistic fermions

2012

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We consider a 1-parameter family of self-adjoint extensions of the Hamiltonian for a particle confined to a finite interval with perfectly reflecting boundary conditions. In some cases, one obtains negative energy states which seems to violate the Heisenberg uncertainty relation. We use this as a motivation to derive a generalized uncertainty relation valid for an arbitrarily shaped quantum dot with general perfectly reflecting walls in d dimensions. In addition, a general uncertainty relation for non-Hermitean operators is derived and applied to the non-Hermitean momentum operator in a quantum dot. We also consider minimal uncertainty wave packets in this situation, and we prove that the spectrum depends monotonically on the self-adjoint extension parameter. In addition, we construct the most general boundary conditions for semiconductor heterostructures such as quantum dots, quantum wires, and quantum wells, which are characterized by a 4-parameter family of selfadjoint extensions. Finally, we consider perfectly reflecting boundary conditions for relativistic fermions confined to a finite volume or localized on a domain wall, which are characterized by a 1-parameter family of self-adjoint extensions in the (1 + 1)-d and (2 + 1)-d cases, and by a 4-parameter family in the (3 + 1)-d and (4 + 1)-d cases.

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Section: Generalized Uncertainty Relationmentioning

confidence: 99%

“…This is the energy-momentum dispersion relation of a fermion moving with the speed 12) and coupled to a chemical potential…”

Section: Domain Wall Boundary Conditions In (2 + 1)-dmentioning

confidence: 99%

From a particle in a box to the uncertainty relation in a quantum dot and to reflecting walls for relativistic fermions

2012

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“…We recall that unique continuation principles for the nonlinear Schrödinger equation and the generalized Korteweg-de Vries equation have been established in [9] and [10] resp. under assumptions on the solutions at two different times.…”

Section: Remarksmentioning

confidence: 99%

The IVP for the Benjamin–Ono equation in weighted Sobolev spaces

Fonseca

Ponce

2011

Journal of Functional Analysis

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102

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We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces Z s,r = H s (R) ∩ L 2 (|x| 2r dx), s ∈ R, s 1 and s r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp.

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“…In this paper we continue our study initiated in [5] [6], and [7] on unique continuation properties of solutions of Schrödinger equations of the form (1.1)…”

Section: Introductionmentioning

confidence: 88%

“…More precisely, in [7] it was shown that there exist (complex-valued) bounded potentials V (x, t) satisfying (1.7) for which there exist nontrivial solutions u ∈ C([0, T ] :…”

Section: Introductionmentioning

confidence: 99%