A Hardy Inequality in Twisted Waveguides

Ekholm, Tomas; Kovařík, Hynek; Krejčiřı́k, David

doi:10.1007/s00205-007-0106-0

Cited by 62 publications

(106 citation statements)

References 16 publications

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“…Section 1.7 The repulsive effect of torsion was first pointed out in [CBr96]. Theorem 1.7 is due to [EKK08], where local versions of Lemma 1.7.2 and Proposition 1.7.1 can also be found. The proof of Lemma 1.7.1 is taken from [BKRS09].…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

146

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

146

133

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Section: Remark 25 (Other Tubes) Assume That L Is Injective and Recmentioning

confidence: 99%

“…The geometrical meaning of our special choice for rotations (R µν ) is that we restrict to non-twisted tubes in the language of [19], which simplifies the analysis considerably. It has been noticed recently in [19,6] that other choices for the rotation may change the spectral picture, too. Namely, in view of the eigenvalue asymptotics obtained in [6], it seems to be reasonable to conjecture that a version of our Theorem 2.3 will still hold for twisted tubes, provided that v 0 is replaced by a more complicated potential, depending also on the higher curvatures of Γ and the geometry of ω.…”

Section: Remark 25 (Other Tubes) Assume That L Is Injective and Recmentioning

confidence: 99%

Location of the nodal set for thin curved tubes

Freitas¹,

Krejčiřı́k²

2008

Indiana Univ. Math. J.

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The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are, in this limit, approximated well by those of the sum of the Dirichlet Laplacian in the cross-section and a one-dimensional Schrödinger operator whose potential is expressed solely in terms of the first curvature of the reference curve. In particular, we establish the convergence of eigenvalues, the uniform convergence of eigenfunctions and locate the nodal set of the Dirichlet Laplacian in the tube near nodal points of the one-dimensional Schrödinger operator. As a consequence, we prove the "nodal-line conjecture" for a class of non-convex and possibly multiply connected domains. The results are based on a perturbation theory developed for Schrödinger-type operators in a straight tube of diminishing cross-section.

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“…Let us renumber the linear array (3) of N = 2K + 1 points in a slightly unusual manner which emphasizes its left-right symmetry,…”

Section: Equivalence To a Linear Discrete Quantum Graphmentioning

confidence: 99%

Fundamental length in quantum theories withPT-symmetric Hamiltonians. II. The case of quantum graphs

Znojil

2009

Phys. Rev. D

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PT −symmetrization of quantum graphs is proposed as an innovation where an adjustable, tunable nonlocality is admitted. The proposal generalizes the PT −symmetric square-well models of Ref.[1] (with real spectrum and with a variable fundamental length θ) which are reclassified as the most elementary quantum q−pointed-star graphs with minimal q = 2. Their equilateral q = 3, 4, . . . generalizations are considered, with interactions attached to the vertices. Runge-Kutta discretization of coordinates simplifies the quantitative analysis by reducing our graphs to star-shaped lattices of N = qK + 1 points. The resulting bound-state spectra are found real in an N−independent interval of couplings λ ∈ (−1, 1). Inside this interval the set of closed-form metrics Θ (N ) j (λ) is constructed, defining independent eligible local (at j = 0) or increasingly nonlocal (at j = 1, 2, . . .) inner products in the respective physical Hilbert spaces of states H (N ) j (λ). In this way each graph is assigned a menu of non-equivalent, optional probabilistic quantum interpretations.

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A Hardy Inequality in Twisted Waveguides

Cited by 62 publications

References 16 publications

Quantum Waveguides

Quantum Waveguides

Location of the nodal set for thin curved tubes

Fundamental length in quantum theories withPT-symmetric Hamiltonians. II. The case of quantum graphs

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