Unique continuation for the magnetic Schrödinger equation

Laestadius, Andre; Benedicks, Michael; Penz, Markus

doi:10.1002/qua.26149

Cited by 6 publications

(9 citation statements)

References 35 publications

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“…This was used by Kurata in [27]to show strong UCP results for oneparticle magnetic Schrödinger operators. Recently, Laestadius, Benedicks and Penz [31] proved the first strong UCP result for many-body magnetic Schrödinger operators, using the work of Kurata. However, they need extra assumptions on (2V + x · V ) − and curl A, and a result with only L p hypothesis on potentials was lacking.…”

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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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confidence: 99%

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

Garrigue

2018

Math Phys Anal Geom

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“…Generally, the unique continuation property (UCP) for solutions of the Schrödinger equation gives conditions on the involved potentials such that if a (distributional) solution vanishes on a set of positive measures, it must vanish everywhere. The question if the UCP holds when the effect of a magnetic field is taken into account was studied on numerous occasions. − The best result for a Hamiltonian of the type of eq was established in Garrigue, and we will repeat it here. The restrictions on the involved potentials is in the form of L loc p spaces on the space domain double-struckR 3 (the reference gives the more general double-struckR d but our treatment is for simplicity restricted to double-struckR 3 ).…”

Section: The Unique Continuation Property For Magnetic Hamiltoniansmentioning

confidence: 99%

The Structure of the Density-Potential Mapping. Part II: Including Magnetic Fields

Penz,

Tellgren,

Csirik

et al. 2023

ACS Phys. Chem Au

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The Hohenberg−Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just onebody particle density. In this Part II of a series of two articles, we aim at clarifying the status of this theorem within different extensions of DFT including magnetic fields. We will in particular discuss currentdensity-functional theory (CDFT) and review the different formulations known in the literature, including the conventional paramagnetic CDFT and some nonstandard alternatives. For the former, it is known that the Hohenberg−Kohn theorem is no longer valid due to counterexamples. Nonetheless, paramagnetic CDFT has the mathematical framework closest to standard DFT and, just like in standard DFT, nondifferentiability of the density functional can be mitigated through Moreau−Yosida regularization. Interesting insights can be drawn from both Maxwell−Schrodinger DFT and quantum-electrodynamic DFT, which are also discussed here.

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“…The former would establish a Hohenberg-Kohn-type mapping, since then (ρ, j) determines (ρ, A) up to a gauge. In a next step one could use the B-DFT extension of the Hohenberg-Kohn theorem to determine v [6,16,26]. In Diener's work, this is in fact the primary intended use of the minimization principle that defines F D .…”

Section: Proposition 2 For Some (ρ A) We Have a Strict Inequality G(ρ...mentioning

confidence: 99%

Revisiting density-functional theory of the total current density

Laestadius

Penz

Tellgren

2021

J. Phys.: Condens. Matter

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Density-functional theory (DFT) requires an extra variable besides the electron density in order to properly incorporate magnetic-field effects. In a time-dependent setting, the gauge-invariant, total current density takes that role. A peculiar feature of the static ground-state setting is, however, that the gauge-dependent paramagnetic current density appears as the additional variable instead. An alternative, exact reformulation in terms of the total current density has long been sought but to date a work by Diener is the only available candidate. In that work, an unorthodox variational principle was used to establish a ground-state DFT of the total current density as well as an accompanying Hohenberg–Kohn-like result. We here reinterpret and clarify Diener’s formulation based on a maximin variational principle. Using simple facts about convexity implied by the resulting variational expressions, we prove that Diener’s formulation is unfortunately not capable of reproducing the correct ground-state energy and, furthermore, that the suggested construction of a Hohenberg–Kohn map contains an irreparable mistake.

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Unique continuation for the magnetic Schrödinger equation

Cited by 6 publications

References 35 publications

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

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

The Structure of the Density-Potential Mapping. Part II: Including Magnetic Fields

Revisiting density-functional theory of the total current density

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