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
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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