%0 Journal Article %T Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry %A Kwang C. Shin %J Symmetry, Integrability and Geometry : Methods and Applications %D 2010 %I National Academy of Science of Ukraine %X We study the eigenvalue problem u''+V(z)u=¦Ëu 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= ¦Ð/2¡À 2¦Ð(m+2), where V(z)= (iz)^m P(iz) for complex-valued polynomials P of degree at most m 1¡Ý2. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues. %K anharmonic oscillators %K asymptotic formula %K infinitely many real eigenvalues %K PT-symmetry %U http://dx.doi.org/10.3842/SIGMA.2010.015