The quasi-static plasmonic problem for polyhedra
Abstract: We characterize the essential spectrum of the plasmonic problem for polyhedra in $${\mathbb {R}}^3$$ R 3 . The description is particularly simple for convex polyhedra and permittivities $$\epsilon < - 1$$ ϵ < - 1 . The plasmonic problem is interpreted as a spectral… Show more
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Cited by 4 publications
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On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains
We say that Γ, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each x ∈ Γ, Γ is either locally C 1 or locally coincides (in some coordinate system centred at x) with a Lipschitz graph Γx such that Γx = αxΓx, for some αx ∈ (0, 1). In this paper we study, for such Γ, the essential spectrum of DΓ, the double-layer (or Neumann-Poincaré) operator of potential theory, on L 2 (Γ). We show, via localisation and Floquet-Bloch-type arguments, that this essential spectrum is the union of the spectra of related continuous families of operators Kt, for t ∈ [−π, π]; moreover, each Kt is compact if Γ is C 1 except at finitely many points. For the 2D case where, additionally, Γ is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of DΓ; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nyström-method approximations to the operators Kt. Through error estimates with explicit constants, we also construct functionals that determine whether any particular locally-dilation-invariant piecewise-analytic Γ satisfies the well-known spectral radius conjecture, that the essential spectral radius of DΓ on L 2 (Γ) is < 1/2 for all Lipschitz Γ. We illustrate this theory with examples; for each we show that the essential spectral radius is < 1/2, providing additional support for the conjecture. We also, via new results on the invariance of the essential spectral radius under locally-conformal C 1,β diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.
On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains
We say that Γ, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each x ∈ Γ, Γ is either locally C 1 or locally coincides (in some coordinate system centred at x) with a Lipschitz graph Γx such that Γx = αxΓx, for some αx ∈ (0, 1). In this paper we study, for such Γ, the essential spectrum of DΓ, the double-layer (or Neumann-Poincaré) operator of potential theory, on L 2 (Γ). We show, via localisation and Floquet-Bloch-type arguments, that this essential spectrum is the union of the spectra of related continuous families of operators Kt, for t ∈ [−π, π]; moreover, each Kt is compact if Γ is C 1 except at finitely many points. For the 2D case where, additionally, Γ is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of DΓ; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nyström-method approximations to the operators Kt. Through error estimates with explicit constants, we also construct functionals that determine whether any particular locally-dilation-invariant piecewise-analytic Γ satisfies the well-known spectral radius conjecture, that the essential spectral radius of DΓ on L 2 (Γ) is < 1/2 for all Lipschitz Γ. We illustrate this theory with examples; for each we show that the essential spectral radius is < 1/2, providing additional support for the conjecture. We also, via new results on the invariance of the essential spectral radius under locally-conformal C 1,β diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.
It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm $$<1/2$$ < 1 / 2 as an operator on the natural trace space $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) whenever $$\Gamma $$ Γ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) for any sequence of finite-dimensional subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) . Long-standing open questions are whether the essential norm is also $$<1/2$$ < 1 / 2 for D as an operator on $$L^2(\Gamma )$$ L 2 ( Γ ) for all Lipschitz $$\Gamma $$ Γ in 2-d; or whether, for all Lipschitz $$\Gamma $$ Γ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators $$\pm \frac{1}{2}I+D$$ ± 1 2 I + D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$L^2(\Gamma )$$ L 2 ( Γ ) . We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is $$\ge 1/2$$ ≥ 1 / 2 , and examples with Lipschitz constant two for which the operators $$\pm \frac{1}{2}I +D$$ ± 1 2 I + D are not coercive plus compact. We also give, for every $$C>0$$ C > 0 , examples of Lipschitz polyhedra for which the essential norm is $$\ge C$$ ≥ C and for which $$\lambda I+D$$ λ I + D is not a compact perturbation of a coercive operator for any real or complex $$\lambda $$ λ with $$|\lambda |\le C$$ | λ | ≤ C . We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the $$L^2(\Gamma )$$ L 2 ( Γ ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on $$C(\Gamma )$$ C ( Γ ) , equivalent to the standard supremum norm, for which the essential norm of D on $$C(\Gamma )$$ C ( Γ ) is $$<1/2$$ < 1 / 2 .
On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains
We say that $$\Gamma $$ Γ , the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each $$x\in \Gamma $$ x ∈ Γ , $$\Gamma $$ Γ is either locally $$C^1$$ C 1 or locally coincides (in some coordinate system centred at x) with a Lipschitz graph $$\Gamma _x$$ Γ x such that $$\Gamma _x=\alpha _x\Gamma _x$$ Γ x = α x Γ x , for some $$\alpha _x\in (0,1)$$ α x ∈ ( 0 , 1 ) . In this paper we study, for such $$\Gamma $$ Γ , the essential spectrum of $$D_\Gamma $$ D Γ , the double-layer (or Neumann–Poincaré) operator of potential theory, on $$L^2(\Gamma )$$ L 2 ( Γ ) . We show, via localisation and Floquet–Bloch-type arguments, that this essential spectrum is the union of the spectra of related continuous families of operators $$K_t$$ K t , for $$t\in [-\pi ,\pi ]$$ t ∈ [ - π , π ] ; moreover, each $$K_t$$ K t is compact if $$\Gamma $$ Γ is $$C^1$$ C 1 except at finitely many points. For the 2D case where, additionally, $$\Gamma $$ Γ is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of $$D_\Gamma $$ D Γ ; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nyström-method approximations to the operators $$K_t$$ K t . Through error estimates with explicit constants, we also construct functionals that determine whether any particular locally-dilation-invariant piecewise-analytic $$\Gamma $$ Γ satisfies the well-known spectral radius conjecture, that the essential spectral radius of $$D_\Gamma $$ D Γ on $$L^2(\Gamma )$$ L 2 ( Γ ) is $$<1/2$$ < 1 / 2 for all Lipschitz $$\Gamma $$ Γ . We illustrate this theory with examples; for each we show that the essential spectral radius is$$<1/2$$ < 1 / 2 , providing additional support for the conjecture. We also, via new results on the invariance of the essential spectral radius under locally-conformal $$C^{1,\beta }$$ C 1 , β diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.
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