Application of the perturbation method of Dingle and M�ller ( applied originally to the Mathieu equation, i . e . a Schr�dinger equation with cosine potential ) requires exploitation of parameter symmetries of the Schr�dinger equation for the quartic potential.
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This gives flexibility in the choice of basis functions for the solution and the linear inversion operators ( as compared to the Adomian decomposition method ), while still retaining a simplicity that makes the method easily understandable from the standpoint of general perturbation methods.
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It is a notable quantum system to solve for; since the solutions are exact ( but complicated in terms of Hermite polynomials ), and it can describe or at least approximate a wide variety of other systems, including basis of perturbation methods in quantum mechanics.
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Bartlett and his coworkers were the first to formulate and implement coupled cluster theory with all single and double excitation operators ( CCSD ) in 1982, followed by triple ( CCSDT ) in 1987, quadruple ( CCSDTQ ), and even quintuple ( CCSDTQP ) excitation operators, and also many-body perturbation methods up to the sixth order.
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An even more convincing calculation is the derivation of Regge trajectories ( and hence eigenvalues ) of the radial Schr�dinger equation with Yukawa potential by both a perturbation method ( with the old l ( l + 1 ) factor ) and independently the derivation by the WKB method ( with Langer replacement )-- in both cases even to higher orders.
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