On the inverse spectral problem for Euclidean triangles

Antunes, Pedro R. S.; Freitas, Pedro

doi:10.1098/rspa.2010.0540

Cited by 12 publications

(11 citation statements)

References 23 publications

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“…In fact, we expect that it is possible to distinguish triangles based on a finite number of eigenvalues. This is supported by numerical data in [3], which shows that one expects that triangles are uniquely determined by their first three eigenvalues. Our work is organized as follows.…”

Section: Introductionsupporting

confidence: 52%

The Fundamental Gap of Simplices

Rowlett

2013

Commun. Math. Phys.

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The gap function of a domain Ω ⊂ R n is ξ(Ω) := d 2 (λ 2 − λ 1 ), where d is the diameter of Ω, and λ 1 and λ 2 are the first two positive Dirichlet eigenvalues of the Euclidean Laplacian on Ω. It was recently shown by Andrews and Clutterbuck [1] that for any convex Ω ⊂ R n ,where the infimum occurs for n = 1. On the other hand, the gap function on the moduli space of n-simplices behaves differently. Our first theorem is a compactness result for the gap function on the moduli space of n-simplices. Next, specializing to n = 2, our second main result proves the recent conjecture of Antunes-Freitas [2]: for any triangle T ⊂ R 2 ,with equality if and only if T is equilateral.arXiv:1109.4117v2 [math.SP]

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Section: Introductionsupporting

confidence: 52%

The Fundamental Gap of Simplices

Rowlett

2013

Commun. Math. Phys.

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“…Both proofs require the entire set of eigenvalues, but it is natural to speculate that the first three eigenvalues are sufficient to determine whether or not two triangles have the same shape. Antunes and Freitas provided strong numerical evidence for this conjecture [3].…”

Section: Motivation and Resultsmentioning

confidence: 86%

The Fundamental Gap and One-Dimensional Collapse

Lu¹,

Rowlett²

2014

Geometric and Spectral Analysis

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Our main result is that if a generic convex domain in R n collapses to a domain in R n−1 , then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate the general case, we first prove the analogous result for triangular and polygonal domains. In so doing, we prove that the first two eigenvalues of triangular domains cannot be polyhomogeneous on the moduli space of triangles without blowing up a certain point. Our results show that the gap generically diverges under one dimensional collapse and is bounded only if the domain is sufficiently close to a rectangle in two dimensions or a cylinder in higher dimensions. 11 3. Collapsing polygonal domains 15 3.1. Proof of Theorem 4: bounded gap 15 3.2. Proof of Theorem 4: unbounded gap 16 4. The general case 19 References 21

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confidence: 67%

“…Antunes and Freitas [1] had observed numerically the presence of a saddle point for λ 4 /λ 1 around which λ 2 /λ 1 is regular, which would imply the existence of such two triangles (see Figure 1). Moreover they conjectured that the three first eigenvalues λ 1 , λ 2 and λ 3 do determine the shape of a triangle.…”

Section: Introductionmentioning

confidence: 93%

Any three eigenvalues do not determine a triangle

Gómez-Serrano¹,

Orriols²

2019

Preprint

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Despite the moduli space of triangles being three dimensional, we prove the existence of two triangles which are not isometric to each other for which the first, second and fourth Dirichlet eigenvalues coincide, establishing a numerical observation from Antunes-Freitas [1]. The two triangles are far from any known, explicit cases. To do so, we develop new tools to rigorously enclose eigenvalues to a very high precision, as well as their position in the spectrum. This result is also mentioned as (the negative) part of [35, Conjecture 6.46], [23, Open Problem 1] and [39, Conjecture 3].

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On the inverse spectral problem for Euclidean triangles

Cited by 12 publications

References 23 publications

The Fundamental Gap of Simplices

The Fundamental Gap of Simplices

The Fundamental Gap and One-Dimensional Collapse

Any three eigenvalues do not determine a triangle

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