Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

We study the eigenvalue problem u''+V(z)u=λu 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.