Bound State Asymptotics for Elliptic Operators with Strongly Degenerated Symbols

Лаптев, Ари; Safronov, Oleg; Weidl, Timo

doi:10.1007/978-1-4615-0777-2_14

Cited by 19 publications

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“…By the definition of B Γ in (5.2), we have Γ at Λ, we need to generalize the well-known weak Neumann-Schatten classes Σ p ; see [24, 11.6], and the classes S p introduced by Safronov [25,17]. This is done as in [15].…”

Section: Lemma 52 For Any V ∈ H 4 the Function B γ V Is A Unique mentioning

confidence: 99%

“…We will need the following two lemmas, which were proved in [15]. See also [17] for the polynomial case.…”

Section: Lemma 52 For Any V ∈ H 4 the Function B γ V Is A Unique mentioning

confidence: 99%

“…The structure of the spectral minimum of a scattering channel is described by the behavior of an eigenvalue branch of a certain effective matrix-valued Sturm-Liouville problem. It turns out that for the scattering channel of interest this minimum is strongly degenerate, what leads to a weak-coupling analysis of a pseudodifferential operator with degenerate symbol in the spirit of [17]. In particular, the minimum will be attained on the circle {ξ ∈ R 2 : |ξ| = κ} for some κ > 0.…”

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“…Furthermore, we state a special type of Birman-Schwinger principle appropriate to our problem. Finally, we introduce the calculus of [17] and describe the behavior of the Birman-Schwinger operator along the spectral minimum of the unperturbed elasticity operator.…”

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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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Section: Lemma 52 For Any V ∈ H 4 the Function B γ V Is A Unique mentioning

confidence: 99%

“…We will need the following two lemmas, which were proved in [15]. See also [17] for the polynomial case.…”

Section: Lemma 52 For Any V ∈ H 4 the Function B γ V Is A Unique mentioning

confidence: 99%

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