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On the reality of the eigenvalues for a class of \(\mathcal{PT}\)-symmetric oscillators. (English) Zbl 1017.34083
Here, the author studies the eigenvalue problem for the equation \[ -u''(z)-[(iz)^{m}+P(iz)]u(z) = \lambda u(z) \] with the boundary conditions that \(u(z)\) decays to zero as \(z\) tends to infinity along the rays \(\arg z = -\frac{\pi}{2} \pm \frac{2\pi}{m+2}\), where \(P(z) = a_{1}z^{m-1}+a_{2}z^{m-2}+...+a_{m-1}z\) is a real polynomial and \(m\geq 2\). It is proved that, if for some \(1\leq j\leq\frac{m}{2}\) one has \((j-k)a_{k}\geq 0\) for all \(1\leq k\leq m-1\), then the eigenvalues are all positive real.

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34M45 Ordinary differential equations on complex manifolds
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