Complex Eigenvalue Splitting for the Dirac Operator
(2021) In Communications in Mathematical Physics 383(3). p.1527-1558- Abstract
We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov–Shabat) operator on the real line with general analytic potential. We provide Bohr–Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd... (More)
We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov–Shabat) operator on the real line with general analytic potential. We provide Bohr–Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd potentials it is horizontal, meaning that all such eigenvalues are purely imaginary when the potential is even, and no such eigenvalue is purely imaginary when the potential is odd.
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- author
- Hirota, Koki and Wittsten, Jens LU
- organization
- publishing date
- 2021-05-01
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Communications in Mathematical Physics
- volume
- 383
- issue
- 3
- pages
- 32 pages
- publisher
- Springer
- external identifiers
-
- scopus:85103665599
- ISSN
- 0010-3616
- DOI
- 10.1007/s00220-021-04063-5
- language
- English
- LU publication?
- yes
- id
- 73742edc-9d99-4ec8-857b-f001b9a60201
- date added to LUP
- 2021-04-14 08:18:28
- date last changed
- 2022-04-27 01:30:48
@article{73742edc-9d99-4ec8-857b-f001b9a60201,
abstract = {{<p>We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov–Shabat) operator on the real line with general analytic potential. We provide Bohr–Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd potentials it is horizontal, meaning that all such eigenvalues are purely imaginary when the potential is even, and no such eigenvalue is purely imaginary when the potential is odd.</p>}},
author = {{Hirota, Koki and Wittsten, Jens}},
issn = {{0010-3616}},
language = {{eng}},
month = {{05}},
number = {{3}},
pages = {{1527--1558}},
publisher = {{Springer}},
series = {{Communications in Mathematical Physics}},
title = {{Complex Eigenvalue Splitting for the Dirac Operator}},
url = {{http://dx.doi.org/10.1007/s00220-021-04063-5}},
doi = {{10.1007/s00220-021-04063-5}},
volume = {{383}},
year = {{2021}},
}