Spectral rigidity for spherically symmetric manifolds with boundary

Hoop, Maarten V. de; Ilmavirta, Joonas; Katsnelson, Vitaly

doi:10.1016/j.matpur.2021.12.009

Cited by 7 publications

(26 citation statements)

References 43 publications

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“…The results presented, here, are an extension of our previous result [4] where we proved a spectral rigidity for a smooth metric on a radial manifold. Allowing for certain discontinuities in the metric adds a new level of challenge for several reasons.…”

Section: Introductionsupporting

confidence: 76%

“…We view M as a Riemannian manifold with (rough) metric g = c −2 dx 2 . We showed in [4] that any rotation symmetric Riemannian manifold with the Herglotz condition is of this form. The same is true in the presence of jumps with essentially the same proof we used in the smooth setting.…”

Section: Unraveling Assumptionsmentioning

confidence: 98%

“…There are several geometric assumptions we make that we shall enumerate here: (A1) "Periodic conjugacy condition." This is an analog of the clean intersection hypothesis used in [4,12,13]; (see Definition 2.5). (A2) "Principal amplitude injectivity condition."…”

Section: Herglotz and Other Conditionsmentioning

confidence: 99%

See 2 more Smart Citations

Spherically symmetric terrestrial planets with discontinuities are spectrally rigid

Ilmavirta¹,

Hoop²,

Katsnelson³

2023

Preprint

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We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in between the interfaces enabling the consideration of two wave types, like P -and S -polarized waves in isotropic elastic solids. Terrestrial planets in our solar system are approximately spherically symmetric and support toroidal and spheroidal modes. Discontinuities typically correspond with phase transitions in their interiors. Our rigidity result applies to such planets as we ensure that our conditions are satisfied in generally accepted models in the presence of a fluid outer core. The proof is based on a novel trace formula.

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Section: Introductionsupporting

confidence: 76%

Section: Unraveling Assumptionsmentioning

confidence: 98%

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Spherically symmetric terrestrial planets with discontinuities are spectrally rigid

Ilmavirta¹,

Hoop²,

Katsnelson³

2023

Preprint

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“…In this case, the solution of the problem reduces to the inversion of an Abel transform [38,51]. There are also recent spectral rigidity results for radial sound speeds which satisfy the Herglotz condition [19].…”

Section: Related Resultsmentioning

confidence: 99%

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

Ilmavirta

Mönkkönen

2023

J Geom Anal

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We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.

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“…In this case the solution of the problem reduces to the inversion of an Abel transform [38,50]. There are also recent spectral rigidity results for radial sound speeds which satisfy the Herglotz condition [14].…”

Section: Introductionmentioning

confidence: 99%

The geodesic ray transform on spherically symmetric reversible Finsler manifolds

Ilmavirta¹,

Mönkkönen²

2022

Preprint

Self Cite

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Spectral rigidity for spherically symmetric manifolds with boundary

Cited by 7 publications

References 43 publications

Spherically symmetric terrestrial planets with discontinuities are spectrally rigid

Spherically symmetric terrestrial planets with discontinuities are spectrally rigid

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

The geodesic ray transform on spherically symmetric reversible Finsler manifolds

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