Quartic anharmonic oscillator with potential V(x)= x² + g²x¿ was the first non-exactly-solvable problem tackled by the newly-written Schrödinger equation in 1926. Since that time thousands of articles have been published on the subject, mostly about the domain of small g² (weak coupling regime), although physics corresponds to g² ~ 1, and they were mostly about energies.


This book is focused on studying eigenfunctions as a primary object for any g². Perturbation theory in g² for the logarithm of the wavefunction is matched to the true semiclassical expansion in powers of ¿: it leads to locally-highly-accurate, uniform approximation valid for any g²¿[0, ¿) for eigenfunctions and even more accurate results for eigenvalues. This method of matching can be easily extended to the general anharmonic oscillator as well as to the radial oscillators. Quartic, sextic and cubic (for radial case) oscillators are considered in detail as well as quartic double-well potential.