Asymptotic relation for zeros of cross-product of Bessel functions and applications

Bobkov, Vladimír

doi:10.1016/j.jmaa.2018.11.065

Cited by 9 publications

(3 citation statements)

References 19 publications

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“…Even though physicists and mathematicians have been dealing with the roots of the equations of the type from Eq. ( 36) since, at least, the end of the 19th century 56 with a lot of knowledge being accumulated over the years 55,[57][58][59][60][61][62][63][64][65][66][67][68] , nowadays its miscellaneous properties continue to attract the researchers 69,70 . Generally, it can be handled numerically only but in the extreme regimes of the thick and thin rings its analytic asymptotic solutions are:…”

Section: Model and Formulationmentioning

confidence: 99%

Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field

Olendski

2022

Preprint

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Shannon quantum information entropies S ρ,γ (φ AB , r 0 ), Fisher informations I ρ,γ (φ AB , r 0 ), Onicescu energies O ρ,γ (φ AB , r 0 ) and Rényi entropies R ρ,γ (φ AB , r 0 ; α) are calculated both in the position (subscript ρ) and momentum (γ) spaces as functions of the inner radius r 0 for the two-dimensional Dirichlet unit-width annulus threaded by the Aharonov-Bohm (AB) flux φ AB . Small (huge) values of r 0 correspond to the thick (thin) rings with extreme of r 0 = 0 describing the dot. Discussion is based on the analysis of the corresponding position Ψ nm (φ AB , r 0 ; r) and momentum Φ nm (φ AB , r 0 ; k) waveforms, with n and m being principal and magnetic quantum indices, respectively: the former allows an analytic expression at any AB field whereas for the latter it is true at the flux-free configuration, φ AB = 0, only. It is shown, in particular, that the position Shannon entropy S ρnm (φ AB , r 0 ) [Onicescu energy O ρnm (φ AB , r 0 )] grows logarithmically [decreases as 1/r 0 ] with large r 0 tending to the same asymptote S asym ρ = ln(4πr 0 ) − 1 [O asym ρ = 3/(4πr 0 )] for all orbitals whereas their Fisher counterpart I ρnm (φ AB , r 0 ) approaches in the same regime the m-independent limit mimicking in this way the energy spectrum variation with r 0 , which for the thin structures exhibits quadratic dependence on the principal index. Frequency of the fading oscillations of the radial parts of the wave vector functions Φ nm (φ AB , r 0 ; k) increases with the inner radius what results in the identical r 0 ≫ 1 asymptote for all momentum Shannon entropies S γnm (φ AB ; r 0 ) with the alike n and different m. The same limit causes the Fisher momentum components I γ (φ AB , r 0 ) to grow exponentially with r 0 . Based on these calculations, properties of the complexities e S O are addressed too. Among many findings on the Rényi entropy, it is proved that the lower limit α T H of the semi-infinite range of the dimensionless coefficient α, where the momentum component of this one-parameter entropy exists, is not influenced by the radius; in particular, the change of the topology from the simply, r 0 = 0, to the doubly, r 0 > 0, connected domain is unable to change α T H = 2/5. AB field influence on the measures is calculated too. Parallels are drawn to the geometry with volcano-shape confining potential and similarities and differences between them are discussed.

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Section: Model and Formulationmentioning

confidence: 99%

Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field

Olendski

2022

Preprint

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“…There are a lot of literature on the study of the zeros of cross-product of Bessel functions. Just to mention a few, M. Kline [11], D. Willis [24], J. Cochran [4,5], V. Bobkov [2], etc. In this section we study such objects from our own perspectives (motivated by the work in [6]), via whose study we look for a two-term Weyl formula for planar annuli.…”

Section: Zeros Of Cross-product Of Bessel Functionsmentioning

confidence: 99%

The Weyl formula for planar annuli

Guo

Müller

Wang

et al. 2019

Preprint

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We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in R 2 . Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amount and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size O(µ 2/3 ). If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely O(µ 131/208 (log µ) 18627/8320 ). As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

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“…Here, κ > 0 is the thickness parameter. The zeros of these functions received plenty of attention with a view towards waveguides [4,5]; they are also used to compute the Pleijel constant for planar annuli [6]. Note that we are being ambiguous saying "zeros": we mean z-zeros while there are also ν-zeros that are of interest [7,8].…”

Section: Introductionmentioning

confidence: 99%

Zeros of Bessel cross-products coming from oblique derivative boundary value problems

Budzinskiy¹

2019

Preprint

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The paper is devoted to (combinations of) Bessel cross-products that arise from oblique derivative boundary value problems for the Laplacian in a circular annulus. We show that like their Neumann-Laplacian counterpart (and unlike the Dirichlet-Laplacian), they possess two kinds of zeros: those that can be derived by McMahon series and diverge to infinity in the limit, and exceptional ones that remain finite. For both cases we find asymptotic expressions for a fixed oblique angle and vanishing thickness of the annulus. We further present plots of numerically computed zeros and discuss their behaviour when the oblique angle changes and the thickness remains fixed.

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Asymptotic relation for zeros of cross-product of Bessel functions and applications

Cited by 9 publications

References 19 publications

Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field

Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field

The Weyl formula for planar annuli

Zeros of Bessel cross-products coming from oblique derivative boundary value problems

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