Bounds on eigenvalues of perturbed Lamé operators with complex potentials

Lucrezia Cossetti; Lucrezia Cossetti

doi:10.3934/mine.2022037

Mathematics in Engineering

2022, Volume 4, Issue 5: 1-29. doi: 10.3934/mine.2022037

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Bounds on eigenvalues of perturbed Lamé operators with complex potentials

Lucrezia Cossetti ^,

Karlsruher Institut für Technologie, Englerstraße 2, 76131 Karlsruhe, Germany

Received: 06 August 2021 Accepted: 20 September 2021 Published: 28 September 2021

Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity $ -\Delta^\ast + V $ in terms of $ L^p $-norms of the potential. Original results within the self-adjoint framework are provided too.
- Lamé operators,
- non self-adjoint operators,
- spectral theory,
- eigenvalue bounds,
- Birman-Schwinger principle
Citation: Lucrezia Cossetti. Bounds on eigenvalues of perturbed Lamé operators with complex potentials[J]. Mathematics in Engineering, 2022, 4(5): 1-29. doi: 10.3934/mine.2022037

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Abstract

Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity $ -\Delta^\ast + V $ in terms of $ L^p $-norms of the potential. Original results within the self-adjoint framework are provided too.

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