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We carry out an exact quantization of a PT symmetric (reversible) Li\'{e}nardtype one dimensional nonlinear oscillator both semiclassically and quantummechanically. The associated time independent classical Hamiltonian is ofnon-standard type and is invariant under a combined coordinate reflection andtime reversal transformation. We use von Roos symmetric ordering procedure towrite down the appropriate quantum Hamiltonian. While the quantum problemcannot be tackled in coordinate space, we show how the problem can besuccessfully solved in momentum space by solving the underlying Schr\"{o}dingerequation therein. We obtain explicitly the eigenvalues and eigenfunctions (inmomentum space) and deduce the remarkable result that the spectrum agreesexactly with that of the linear harmonic oscillator, which is also confirmed bya semiclassical modified Bohr-Sommerfeld quantization rule, while theeigenfunctions are completely different.

Exact quantization of a PT-symmetric (reversible) Li\'enard-type nonlinear oscillator