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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Newton polygons and resonances of multiple delta-potentials
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by Kiril Datchev, Jeremy L. Marzuola and Jared Wunsch;
Trans. Amer. Math. Soc. 377 (2024), 2009-2025
DOI: https://doi.org/10.1090/tran/9056
Published electronically: November 20, 2023

Abstract:

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb {R}$. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form $\operatorname {Im}z \sim -\gamma h \log (1/h).$ More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter $\gamma .$ We present numerical evidence of the existence of more and more possible values of $\gamma$ for larger numbers of delta poles.
References
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Bibliographic Information
  • Kiril Datchev
  • Affiliation: Mathematics Department, Purdue University, West Lafayette, Indiana 47907
  • MR Author ID: 860651
  • Email: kdatchev@purdue.edu
  • Jeremy L. Marzuola
  • Affiliation: Mathematics Department, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599
  • MR Author ID: 787291
  • ORCID: 0000-0002-8668-7143
  • Email: marzuola@math.unc.edu
  • Jared Wunsch
  • Affiliation: Mathematics Department, Northwestern University, Evanston, Illinois 60208
  • MR Author ID: 362611
  • ORCID: 0000-0001-7466-6858
  • Email: jwunsch@math.northwestern.edu
  • Received by editor(s): August 31, 2022
  • Received by editor(s) in revised form: August 8, 2023
  • Published electronically: November 20, 2023
  • Additional Notes: The first author was supported from NSF grant DMS-1708511. The second author was supported from NSF grant DMS-1909035. The third author was partially supported by Simons Foundation grant 631302, NSF grant DMS–2054424, and a Simons Fellowship.
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 377 (2024), 2009-2025
  • MSC (2020): Primary 81U24, 35B34
  • DOI: https://doi.org/10.1090/tran/9056
  • MathSciNet review: 4744748