On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential

Helffer, Bernard; Nourrigat, Jean

doi:10.1007/978-3-030-12661-2_8

Cited by 7 publications

(8 citation statements)

References 28 publications

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“…Setting ≡ 0 in (3.4), our Theorem 3.2 directly yields a maximal inequality for scalar Schrödinger operator with purely imaginary potential which can be compared with the results of [17]. It should be noted that the approach there is very different from ours, resulting in very different assumptions on the function .…”

Section: The Generation Resultsmentioning

confidence: 83%

“…Example We can also extend Example slightly, to show that from our results about vector‐valued Schrödinger operators, we can also infer information about scalar‐valued Schrödinger operators with complex potential. Indeed, given

v, w : {double-struckR}^{d} \to R

, we may consider the matrix potential

V false(x false) : = (\begin{matrix} w false(x false) & - v false(x false) \\ v false(x false) & w false(x false) \end{matrix}) = w false(x false) (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) + v false(x false) (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) .

Diagonalizing the latter matrix via the matrix P from Example , we see that the vector valued operator

Δ + V

is similar, via the same transformation P , to the diagonal operator

(\begin{matrix} Δ & 0 \\ 0 & Δ \end{matrix}) + (\begin{matrix} w false(x false) + i v false(x false) & 0 \\ 0 & w false(x false) - i v false(x false) \end{matrix}) .

Setting

w \equiv 0

in , our Theorem directly yields a maximal inequality for scalar Schrödinger operator with purely imaginary potential which can be compared with the results of . It should be noted that the approach there is very different from ours, resulting in very different assumptions on the function v .…”

Section: The Generation Resultsmentioning

confidence: 99%

“…It should be noted that the approach there is very different from ours, resulting in very different assumptions on the function v . While makes high regularity assumptions (namely that v is

C^{\infty}

), the growth assumptions are not very restrictive.…”

Section: The Generation Resultsmentioning

confidence: 99%

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‐Theory for Schrödinger systems

Kunze

Lorenzi

Maichine

et al. 2019

Mathematische Nachrichten

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In this article, we study for p∈(1,∞) the Lp‐realization of the vector‐valued Schrödinger operator Lu:=prefix div (Q∇u)+Vu. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the Lp‐realization of scriptL, defined on the intersection of the natural domains of the differential and multiplication operators which form scriptL, generates a strongly continuous contraction semigroup on Lp(double-struckRd;double-struckCm). We also study additional properties of the semigroup such as extension to L1, positivity, ultracontractivity and prove that the generator has compact resolvent.

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Section: The Generation Resultsmentioning

confidence: 83%

v, w : {double-struckR}^{d} \to R

, we may consider the matrix potential

V false(x false) : = (\begin{matrix} w false(x false) & - v false(x false) \\ v false(x false) & w false(x false) \end{matrix}) = w false(x false) (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) + v false(x false) (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) .

Diagonalizing the latter matrix via the matrix P from Example , we see that the vector valued operator

Δ + V

is similar, via the same transformation P , to the diagonal operator

(\begin{matrix} Δ & 0 \\ 0 & Δ \end{matrix}) + (\begin{matrix} w false(x false) + i v false(x false) & 0 \\ 0 & w false(x false) - i v false(x false) \end{matrix}) .

Setting

w \equiv 0

Section: The Generation Resultsmentioning

confidence: 99%

See 1 more Smart Citation

‐Theory for Schrödinger systems

Kunze

Lorenzi

Maichine

et al. 2019

Mathematische Nachrichten

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“…Remark. The maximal accretivity of the magnetic Schrödinger operator P in (1.9) holds in great generality, assuming only that (1.4) holds, see [8] and references given there.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Semigroup expansions for non-selfadjoint Schrödinger operators

Bellis

Hitrik

2019

Journal of Functional Analysis

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“…By standard elliptic estimates we then conclude that ũ3 ∈ H 2 (R k ). We can then apply on the two first lines [19,Theorem 5] to conclude that…”

Section: Characterization Of the Domainmentioning

confidence: 99%

On the spectrum of some Bloch-Torrey vector operators

Almog,

Helffer

2020

Preprint

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We consider the Bloch-Torrey operator in L 2 (I, R 3 ) where I ⊆ R. In contrast with the L 2 (I, R 2 ) (as well as the L 2 (R k , R 2 )) case considered in previous works. We obtain that R + is in the continuous spectrum for I = R as well as discrete spectrum outside the real line. For a finite interval we find the left margin of the spectrum. In addition, we prove that the Bloch-Torrey operator must have an essential spectrum for a rather general setup in R k , and find an effective description for its domain.

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On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential

Cited by 7 publications

References 28 publications

‐Theory for Schrödinger systems

‐Theory for Schrödinger systems

Semigroup expansions for non-selfadjoint Schrödinger operators

On the spectrum of some Bloch-Torrey vector operators

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