Piecewise-polynomial approximation of functions fromH ℓ((0, 1) d ), 2ℓ=d, and applications to the spectral theory of the Schrödinger operator

Solomyak, Michael

doi:10.1007/bf02773681

Cited by 61 publications

(126 citation statements)

References 7 publications

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“…Similar estimates, in terms of logarithmic Lieb-Thirring inequalities, for the operator V in dimension two were obtained in [14]. Upper bounds on N. V; 0/ including logarithmic weights were studied in [6], [25], and [27].…”

Section: Proofs Of the Main Results: Radial Fieldsmentioning

confidence: 53%

Eigenvalue bounds for two-dimensional magnetic Schrödinger operators

Kovařík¹

2011

J. Spectr. Theory

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Abstract.We prove that the number of negative eigenvalues of two-dimensional magnetic Schrödinger operators is bounded from above by the strength of the corresponding electric potential. Such estimates fail in the absence of a magnetic field. We also show how the corresponding upper bounds depend on the properties of the magnetic field and discuss their connection with Hardy-type inequalities. Mathematics Subject Classification (2010). 35P05, 35P15.

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Section: Proofs Of the Main Results: Radial Fieldsmentioning

confidence: 53%

Eigenvalue bounds for two-dimensional magnetic Schrödinger operators

Kovařík¹

2011

J. Spectr. Theory

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“…In two dimensions there exist certain CLR-type estimates both for L 0 − V [34,6] and L(A) − V [29], provided A j ∈ L 2 loc (R 2 ) for the latter operator. However, unlike in higher dimensions, these estimates, having a different form, do not produce Lieb-Thirring inequalities.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians

Melgaard

Ouhabaz

Rozenblum

2004

Ann. Henri Poincaré

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The diamagnetic inequality is established for the Schrödinger operator H (d) 0in L 2 (R d ), d = 2, 3, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in R 2 , e.g., a lattice. This fact is used to prove the Lieb-Thirring inequality as well as CLR-type eigenvalue estimates for the perturbed Schrödinger operator H (d) 0 − V , using new Hardy type inequalities. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also proved.

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“…There are some results for the case n = 4 in [3,4,19,20], but the article of greatest relevance to us here is [9] where an upper bound is obtained for N ∆ 2 + c |x| 2 − V (c a positive constant) which coincides with (1.5) when V is radial. In this paper we analyse the spectral properties of perturbations of the magnetic bi-harmonic operator ∆ 2 A , mainly in the cases n = 2, 3, 4.…”

Section: Introductionmentioning

confidence: 89%

Counting eigenvalues of biharmonic operators with magnetic fields

Evans

Lewis

2005

Mathematische Nachrichten

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An analysis is given of the spectral properties of perturbations of the magnetic bi-harmonic operator ∆ 2 A in L 2 (R n ), n = 2, 3, 4, where A is a magnetic vector potential of Aharonov-Bohm type, and bounds for the number of negative eigenvalues are established. Key elements of the proofs are newly derived Rellich inequalities for ∆ 2 A which are shown to have a bearing on the limiting cases of embedding theorems for Sobolev spaces H 2 (R n ).

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Piecewise-polynomial approximation of functions fromH ℓ((0, 1) d ), 2ℓ=d, and applications to the spectral theory of the Schrödinger operator

Cited by 61 publications

References 7 publications

Eigenvalue bounds for two-dimensional magnetic Schrödinger operators

Eigenvalue bounds for two-dimensional magnetic Schrödinger operators

Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians

Counting eigenvalues of biharmonic operators with magnetic fields

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