Any three eigenvalues do not determine a triangle

Gómez-Serrano, Javier; Orriols, G.

doi:10.48550/arxiv.1911.06758

Cited by 2 publications

(2 citation statements)

References 47 publications

(68 reference statements)

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“…When the terms in the expansion are not orthogonal, more work is required. We apply a method used by Gómez-Serrano and Orriols [22] in the context of polygons. The idea is that if u does not vanish in a subset Ω of Ω, without loss of generality it can be assumed to be positive there and then, since −∆u = λu > 0, u is a superharmonic function and satisfies inf Ω u ≥ inf ∂Ω u.…”

Section: 2mentioning

confidence: 99%

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Computation of Tight Enclosures for Laplacian Eigenvalues

Dahne,

Salvy

2020

Preprint

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Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions from all corners and an expansion from an interior point. In particular, this allows us to compute 80 digits of the fundamental eigenvalue for the 3D Kreweras model that has been the object of previous efforts.

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Section: 2mentioning

confidence: 99%

“…Very recently, Gómez-Serrano and Orriols [22] have proved that three eigenvalues do not determine a triangle. For this, they used the method of particular solutions in the plane in a spirit very similar to ours.…”

Section: Introductionmentioning

confidence: 99%

Computation of Tight Enclosures for Laplacian Eigenvalues

Dahne,

Salvy

2020

Preprint

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Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry

Berghaus,

Jones,

Monien

et al. 2024

Adv Comput Math

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We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues $$\lambda (n)$$ λ ( n ) of shapes with n edges that are of the form $$\lambda (n) \sim x\sum _{k=0}^{\infty } \frac{C_k(x)}{n^k}$$ λ ( n ) ∼ x ∑ k = 0 ∞ C k ( x ) n k where x is the limiting eigenvalue for $$n\rightarrow \infty $$ n → ∞ . Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order $$C_k(x)$$ C k ( x ) and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary).

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Any three eigenvalues do not determine a triangle

Cited by 2 publications

References 47 publications

Computation of Tight Enclosures for Laplacian Eigenvalues

Computation of Tight Enclosures for Laplacian Eigenvalues

Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry

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