Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT -Symmetry

[摘要] We study the eigenvalue problem − u ''+ V ( z ) u =λ 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 =−π/2± 2π( m +2), where V ( z )=−( iz ) m − P ( iz ) for complex-valued polynomials P of degree at most m −1≥2. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.

[关键词] anharmonic oscillators;asymptotic formula;infinitely many real eigenvalues;PT -symmetry