Spectral Asymptotics of Pauli Operators and Orthogonal Polynomials in Complex Domains

Filonov, N.; Pushnitski, Alexander

doi:10.1007/s00220-006-1520-0

Cited by 27 publications

(50 citation statements)

References 21 publications

(47 reference statements)

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“…Note that such accumulation rates have recently been obtained also in other physical settings; see, e.g., [18]- [20]. In general, they rest on the combination of compactly supported perturbations applied to strongly degenerate operators.…”

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confidence: 68%

Trapped modes in an elastic plate with a hole

Förster

Weidl

2012

St. Petersburg Math. J.

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Abstract. For an infinite linear elastic plate with stress-free boundary, the trapped modes arising around holes in the plate are investigated. These are L 2 -eigenvalues of the elastostatic operator in the punched plate subject to Neumann type stress-free boundary conditions at the surface of the hole. It is proved that the perturbation gives rise to infinitely many eigenvalues embedded into the essential spectrum. The eigenvalues accumulate to a positive threshold. An estimate of the accumulation rate is given. §1. Introduction We consider a homogeneous and isotropic linear elastic medium in the domainThen the quadratic forminduces the elastostatic operator A for materials with zero Poisson ratio; A corresponds to the differential expressionwith stress-free (Neumann-type) boundary conditions. 1 The operator A has purely absolutely continuous spectrum filling the nonnegative half-line. Now, consider a punched plate Ω c = G \ Ω with a hole Ω = Ω 0 × J, where Ω 0 ⊂ R 2 is a bounded Lipschitz domain. Let A Ω c be the elastostatic operator corresponding to the differential expression (1.2) on the outer domain Ω c subject to stress-free (Neumanntype) boundary conditions. This geometric perturbation does not change the location of the essential spectrum, but it gives rise to a somewhat unexpected trapping effect. As the main result of this paper we prove the existence of infinitely many eigenvalues {ν k } k∈N of A Ω c , embedded into the essential spectrum, which accumulate at a certain threshold Λ > 0, and we compute the accumulation rate of these trapped modes. We establish the formula2010 Mathematics Subject Classification. Primary 74B05. Key words and phrases. Elasticity operator, trapped modes. 1 Here we have chosen a suitable set of units such that Young's modulus E fulfills E = 2. Moreover, we point out that the special choice of zero Poisson ratio is essential for the results of this paper.

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confidence: 68%

Trapped modes in an elastic plate with a hole

Förster

Weidl

2012

St. Petersburg Math. J.

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“…-In the recent article [12] a sharper version of the result of Lemma 5 has been obtained, containing three asymptotic terms as s 0 of n + (s; p q U p q ) provided that the support of U is compact, and U satisfies some additional technical assumptions. In particular, the asymptotic expansion of n + (s; p q U p q ) obtained in [12] recovers the logarithmic capacity of the support of U .…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Resonances and Spectral Shift Function near the Landau levels

Bony¹,

Bruneau²,

Raikov³

2007

Ann. inst. Fourier

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“…Nonetheless, the rest of the article is dedicated to the case where V 2 L 1 0 .R 2 I R/ has a compact support. This choice is motivated by the possible applications in the theory of the quantum Hall effect (see, e.g., [6], [10], and [5]), and, on the other hand, by the spectacular progress in the investigation of the discrete spectrum for localized perturbations of the Landau Hamiltonian H 0 .b; 0/ (see, e.g., [23], [15], [18], [7], [26], [17], and [19]). For definiteness, we suppose that V 0 and discuss only the behavior of the counting functions N C j .…”

Section: Introductionmentioning

confidence: 99%

Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials

Bruneau¹,

Miranda²,

Raikov³

2011

J. Spectr. Theory

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Abstract. We consider the unperturbed operatorHere A is a magnetic potential which generates a constant magnetic field b > 0, and the edge potential W is a non-decreasing non-constant bounded function depending only on the first coordinate x 2 R of .x; y/ 2 R 2 . Then the spectrum of H 0 has a band structure and is absolutely continuous; moreover, the assumption lim x!1 .W .x/ W . x// < 2b implies the existence of infinitely many spectral gaps for H 0 . We consider the perturbed operators H˙D H 0˙V where the electric potential V 2 L 1 .R 2 / is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H˙in the spectral gaps of H 0 . We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to V . Further, we restrict our attention on perturbations V of compact support and constant sign. We establish a geometric condition on the support of V which guarantees the finiteness of the number of the eigenvalues of H˙in any spectral gap of H 0 . In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of H C (resp. H ) to the lower (resp. upper) edge of a given spectral gap, is Gaussian.Mathematics Subject Classification (2010). 35P20, 35J10, 47F05, 81Q10.

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Spectral Asymptotics of Pauli Operators and Orthogonal Polynomials in Complex Domains

Cited by 27 publications

References 21 publications

Trapped modes in an elastic plate with a hole

Trapped modes in an elastic plate with a hole

Resonances and Spectral Shift Function near the Landau levels

Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials

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