Cwikel's theorem and the CLR inequality

Frank, Rupert L.

doi:10.4171/jst/59

Cited by 43 publications

(51 citation statements)

References 23 publications

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“…Analogues of Theorem 3 are widely known for many bounded from below selfadjoint operators as Cwickel-Lieb-Rozenblum inequalities (see [26,6,19] for the original contributions and [12] and references therein for further developments). In particular, in Example 3.3 of [12] it is proved that the estimate…”

Section: Introductionmentioning

confidence: 99%

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

Morozov

Muller

2018

J. Spectr. Theory

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Section: Introductionmentioning

confidence: 99%

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

Morozov

Muller

2018

J. Spectr. Theory

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“…It was conjectured by Simon and proved by Cwikel (see also ) that conditions

f \in L_{p} false(R^{d} false)

and

g \in L_{p, \infty} false(R^{d} false)

p > 2

, ensure that the operator

M_{f} g false(- i \nabla false)

belongs to the weak Schatten ideal

{scriptL}_{p, \infty} false(L_{2} ({double-struckR}^{d}) false)

and the norm of

M_{f} g false(- i \nabla false)

{scriptL}_{p, \infty} false(L_{2} ({double-struckR}^{d}) false)

is dominated by the product of

{false∥ f false∥}_{p}

and

{false∥ g false∥}_{p, \infty}

. Estimates given in Simon's book and deeper results in this direction can be found in .…”

Section: Introductionmentioning

confidence: 99%

Cwikel estimates revisited

Levitina¹,

Sukochev²,

Zanin³

2019

Proc. London Math. Soc.

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In this paper, we extend and strengthen classical estimates for singular values of integral operators originally due to Cwikel, Birman and Solomyak. Suppose that A1 and A2 are two semifinite von Neumann algebras with representations π1:scriptA1→Bfalse(scriptHfalse) and π2:scriptA2→Bfalse(scriptHfalse) such that false∥π1false(xfalse)π2false(yfalse)∥scriptL2false(scriptHfalse)⩽const ∥x∥scriptL2false(A1false)∥y∥scriptL2false(scriptAfalse).Our first result asserts that the inequality false∥π1false(xfalse)π2false(yfalse)∥E(H)⩽CE∥x⊗y∥E(scriptA1⊗scriptA2)holds in any interpolation space E for (L2,L∞). In the special case, when scriptA1=scriptA2=L∞false(Rdfalse) and π1, π2 are given by multipliers and Fourier multipliers, respectively, our result yields a (strengthened version) of well‐known Cwikel estimates [Cwikel, Ann. of Math. (2) 106 (1977) 93–100]. We demonstrate further the applicability of our result by considering noncommutative Euclidean space (Moyal plane) and magnetic Laplacian. Our second direction relates to Birman–Solomyak estimates for interpolation space E for the (quasi)‐Banach couple (ℓp,ℓ2), p<2. In this setting, our technique yields substantial strengthening of results from [Birman and Solomyak, Russian Math. Surveys 32 (1977) 15–89] and Chapter VI of Simon [Trace ideals and their applications (American Mathematical Society, Providence, RI, 2005)]. Finally, we provide Cwikel–Birman–Solomyak estimates for the crucial case of weak Schatten p‐ideals, 1⩽p<2, in the setting of noncommutative Euclidean space.

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“…Hence the order of integrations in (5.2) can be interchanged by Fubini's theorem. By Schwarz inequality and (5.3) we have 5) which is positive for n big enough due to (2.6) and (1.3). It follows from (4.33), (4.34), (4.27) and Lemmata 9 and 10, that there exists h ν,κ > 0 such that for…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…Since R 1 ∈ L ∞ (R + , rdr), there exists α 0 > 0 such that for all κ satisfying (7.1) Now we invoke the Cwikel-Lieb-Rozenblum inequality for (−∆) 1/2 in R 2 : By Remark 2.5 in [2] (or Example 3.3 in [5]) there exists C CLR > 0 such that the right hand side of (7.13) does not exceed…”

Section: Proof Of Theoremmentioning

confidence: 99%

On the Virtual Levels of Positively Projected Massless Coulomb–Dirac Operators

Morozov

Muller

2017

Ann. Henri Poincaré

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Considering different self-adjoint realisations of positively projected massless Coulomb-Dirac operators we find out, under which conditions any negative perturbation, however small, leads to emergence of negative spectrum. We also prove some weighted Lieb-Thirring estimates for negative eigenvalues of such operators. In the process we find explicit spectral representations for all self-adjoint realisations of massless Coulomb-Dirac operators on the half-line.

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Cwikel's theorem and the CLR inequality

Cited by 43 publications

References 23 publications

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

Cwikel estimates revisited

On the Virtual Levels of Positively Projected Massless Coulomb–Dirac Operators

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