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Anharmonic oscillators with infinitely many real eigenvalues and \(\mathcal{PT}\)-symmetry. (English) Zbl 1201.34135

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.

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
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