On the reality of the eigenvalues for a class of \(\mathcal{PT}\)-symmetric oscillators. (English) Zbl 1017.34083

Shin, K. C.

Commun. Math. Phys. 229, No. 3, 543-564 (2002).

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.

Reviewer: Nikolay Vasilye Grigorenko (Kyïv)

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

Keywords:

eigenvalues; \({\mathcal P}{\mathcal T}\)-symmetric oscillators