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Spectral and Angle-Resolved Magneto-Optical Characterization of Photonic Nanostructures

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Bounds for spectral projectors on generic tori.

Pierre Germain¹, Simon L Rydin Myerson²

¹Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185 USA.

Mathematische Annalen

|January 29, 2024

Summary

We study spectral projector norms on thin spherical shells for the Laplacian operator on tori. Our findings improve upon existing results for arbitrary tori.

Keywords:

11D75 11H06 11L07 11P21 42A45 42B15

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Area of Science:

Mathematical analysis
Spectral theory
Differential geometry

Background:

The study of spectral projectors is crucial in understanding the behavior of operators on various manifolds.
Previous research has focused on arbitrary tori, but specific properties on thin spherical shells require further investigation.

Purpose of the Study:

To investigate the norms of spectral projectors on thin spherical shells for the Laplacian operator.
To develop and partially prove a conjecture regarding these norms on generic tori, including rectangular ones.

Main Methods:

The research employs advanced techniques in spectral theory and functional analysis.
Methods involve analyzing the behavior of spectral projectors in the limit of thin shells.

Main Results:

A conjecture concerning the norms of spectral projectors on thin spherical shells is formulated.
Partial proof of the conjecture is provided, offering improved bounds compared to previous results for arbitrary tori.

Conclusions:

The findings contribute to a deeper understanding of spectral properties on manifold with specific geometric constraints.
The results offer a foundation for future research into spectral theory on more complex geometric structures.