The Birkhoff–Gustavson normal form (BGNF) is employed to study general double‐well anharmonic oscillators with polynomial potentials of degree 4. Via an analytic continuation of the BGNF series, expressions are presented that provide classical and semiclassical results for all energies. A surprising observation of the classical period and an interesting feature of the semiclassical quantum numbers are reported. It is shown that except for a small region near the top of the potential barrier, the BGNF approach yields accurate quantum energies.
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