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arXiv · 1008.0905

Anharmonic oscillators in the complex plane, $\mathcal{PT}$-symmetry, and real eigenvalues

Kwang C. Shin
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Abstract

For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problems $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2}$ in the complex plane, where $P$ is a polynomial of degree at most $m-1$. We provide asymptotic expansions of the eigenvalues $\lambda_{n}$. Then we show that if the eigenvalue problem is $\mathcal{PT}$-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when $\gcd(m,\ell)=1$, the eigenvalue problem has infinitely many real eigenvalues if and only if its translation or itself is $\mathcal{PT}$-symmetric. Also, we will prove some other interesting direct and inverse spectral results.

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Kwang C. Shin. 2010-08-05. Anharmonic oscillators in the complex plane, $\mathcal{PT}$-symmetry, and real eigenvalues. https://arxiv.org/abs/1008.0905

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