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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weyl asymptotics for functional difference operators with power to quadratic exponential potential
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by Yaozhong Qiu;
Proc. Amer. Math. Soc. 152 (2024), 3339-3351
DOI: https://doi.org/10.1090/proc/16765
Published electronically: June 14, 2024

Abstract:

We continue the program first initiated by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and develop a modification of the technique introduced in that paper to study the spectral asymptotics, namely the Riesz means and eigenvalue counting functions, of functional difference operators $H_0 = \mathcal {F}^{-1} M_{\cosh (\xi )} \mathcal {F}$ with potentials of the form $W(x) = \left \lvert {x} \right \rvert ^pe^{\left \lvert {x} \right \rvert ^\beta }$ for either $\beta = 0$ and $p > 0$ or $\beta \in (0, 2]$ and $p \geq 0$. We provide a new method for studying general potentials which includes the potentials studied by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and [J. Math. Phys. 60 (2019), p. 103505]. The proof involves dilating the variance of the gaussian defining the coherent state transform in a controlled manner preserving the expected asymptotics.
References
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Bibliographic Information
  • Yaozhong Qiu
  • Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
  • ORCID: 0000-0002-0649-1918
  • Email: y.qiu20@imperial.ac.uk
  • Received by editor(s): May 30, 2023
  • Received by editor(s) in revised form: December 7, 2023, and December 8, 2023
  • Published electronically: June 14, 2024
  • Additional Notes: The author was supported by the President’s Ph.D. Scholarship of Imperial College London
  • Communicated by: Tanya Christiansen
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 3339-3351
  • MSC (2020): Primary 34K08, 34L15
  • DOI: https://doi.org/10.1090/proc/16765
  • MathSciNet review: 4767266