Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry
We study the eigenvalue problem −u''+V(z)u=λu in the complex plane with the boundary condition that u(z) decays to zero as z tends to infinity along the two rays arg z=−π/2± 2π(m+2), where V(z)=−(iz)^m−P(iz) for complex-valued polynomials P of degree at most m−1≥2. We provide an asymptotic...
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National Academy of Science of Ukraine
2010-02-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2010.015 |
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doaj-3e1002ae47884fb69031c99b833475102020-11-24T22:22:59ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592010-02-016015Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-SymmetryKwang C. ShinWe study the eigenvalue problem −u''+V(z)u=λu in the complex plane with the boundary condition that u(z) decays to zero as z tends to infinity along the two rays arg z=−π/2± 2π(m+2), where V(z)=−(iz)^m−P(iz) for complex-valued polynomials P of degree at most m−1≥2. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues. http://dx.doi.org/10.3842/SIGMA.2010.015anharmonic oscillatorsasymptotic formulainfinitely many real eigenvaluesPT-symmetry |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Kwang C. Shin |
| spellingShingle |
Kwang C. Shin Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry Symmetry, Integrability and Geometry: Methods and Applications anharmonic oscillators asymptotic formula infinitely many real eigenvalues PT-symmetry |
| author_facet |
Kwang C. Shin |
| author_sort |
Kwang C. Shin |
| title |
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry |
| title_short |
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry |
| title_full |
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry |
| title_fullStr |
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry |
| title_full_unstemmed |
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry |
| title_sort |
anharmonic oscillators with infinitely many real eigenvalues and pt-symmetry |
| publisher |
National Academy of Science of Ukraine |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| issn |
1815-0659 |
| publishDate |
2010-02-01 |
| description |
We study the eigenvalue problem −u''+V(z)u=λu in the complex plane with the boundary condition that u(z) decays to zero as z tends to infinity along the two rays arg z=−π/2± 2π(m+2), where V(z)=−(iz)^m−P(iz) for complex-valued polynomials P of degree at most m−1≥2. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues. |
| topic |
anharmonic oscillators asymptotic formula infinitely many real eigenvalues PT-symmetry |
| url |
http://dx.doi.org/10.3842/SIGMA.2010.015 |
| work_keys_str_mv |
AT kwangcshin anharmonicoscillatorswithinfinitelymanyrealeigenvaluesandptsymmetry |
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