## 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.

### MSC:

 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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