Lieb-Thirring Inequalities for Geometrically Induced Bound States

Exner, Pavel; Linde, Helmut; Weidl, Timo

doi:10.1007/s11005-004-1741-0

Cited by 28 publications

(11 citation statements)

References 28 publications

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“…A similar decomposition can be used to prove LT-type inequalities also in situations when the perturbation is not given by a potential but rather it is due to deformation and/or boundary conditions-see Problems 5-7 and [ELW04] for a more detailed discussion.…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

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142

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

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142

133

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“…see [LW00], and also [ELM04,Wei08]. The integral at the right-hand side of (2.1), in fact restricted to those x 3 for which inf spec(−∆ ω(x 3 ) ) < Λ, yields the classical phase space volume.…”

Section: Dimensional Reductionmentioning

confidence: 99%

Semiclassical bounds in magnetic bottles

et al. 2016

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The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in $\mathbb{R}^3$ confined by a local change of the magnetic field. We establish two-dimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.Comment: 35 pages, no figure

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“…The proof of Theorem 3.1 relies on a lifting technique, which was introduced in [25], see also [26], [8], [33], and [9] for further developments and applications.…”

Section: Remark 32 Let Us Definementioning

confidence: 99%

Geometrical Versions of improved Berezin–Li–Yau Inequalities

Geisinger¹,

Лаптев²,

Weidl³

2011

J. Spectr. Theory

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Abstract. We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set inIn particular, we derive upper bounds on Riesz means of order 3=2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.Mathematics Subject Classification (2010). Primary 35P15; Secondary 47A75.

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Lieb-Thirring Inequalities for Geometrically Induced Bound States

Cited by 28 publications

References 28 publications

Quantum Waveguides

Quantum Waveguides

Semiclassical bounds in magnetic bottles

Geometrical Versions of improved Berezin–Li–Yau Inequalities

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