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Anharmonic Oscillators with Infinitely Many Real Eigenvalues and $\mathcal{PT}$-Symmetry
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the eigenvalue problem $-u''+V(z)u=\lambda u$ in the complex plane with the boundary condition that $u(z)$ decays to zero as $z$ tends to infinity along the two rays $\arg z=-\frac\pi2\pm \frac2\pi{m+2}$, where $V(z)=-(iz)^m-P(iz)$ for complex-valued polynomials $P$ of degree at most $m-1\ge 2$. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.
Keywords: anharmonic oscillators; asymptotic formula; infinitely many real eigenvalues; $\mathcal{PT}$-symmetry. 
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     author = {Kwang C. Shin},
     title = {Anharmonic {Oscillators} with {Infinitely} {Many} {Real} {Eigenvalues} and $\mathcal{PT}${-Symmetry}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a14/}
}
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Kwang C. Shin. Anharmonic Oscillators with Infinitely Many Real Eigenvalues and $\mathcal{PT}$-Symmetry. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a14/

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