Shin, Kwang C. Anharmonic oscillators with infinitely many real eigenvalues and \(\mathcal{PT}\)-symmetry. (English) Zbl 1201.34135 SIGMA, Symmetry Integrability Geom. Methods Appl. 6, Paper 015, 9 p. (2010). Summary: 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\(z= - \pi /2\pm 2\pi (m+2)\), where \(V(z)= - (iz)m - P(iz)\) for complex-valued polynomials \(P\) of degree at most \(m - 1\geq 2\). We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues. Cited in 2 Documents MSC: 34M03 Linear ordinary differential equations and systems in the complex domain 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators Keywords:anharmonic oscillators; asymptotic formula; infinitely many real eigenvalues; \(\mathcal{PT}\)-symmetry PDFBibTeX XMLCite \textit{K. C. Shin}, SIGMA, Symmetry Integrability Geom. Methods Appl. 6, Paper 015, 9 p. (2010; Zbl 1201.34135) Full Text: DOI arXiv EuDML EMIS