5.4 The coupled equations

The coupled equations (5.8) can be solved by several numerical means. For instance, one can use the discretization algorithm described in Section 2.4 for mesons, which is easily generalized to coupled equations.

Although present computers allow one to diagonalize very large matrices, it seems desirable to optimize the efficiency of the method. The main question deals with the HH one has to actually include in the hyperspherical expansion (5.6). A simple strategy consists of taking all HH of appropriate quantum numbers up to a value L_max of the grand orbital momentum and studying the convergence as L_max increases. In fact, for a given L, only a few HH contribute significantly. This leads to the improved strategy of

potential harmonics [53]. Consider for instance the ground state, whose wave function is scalar and symmetric. Then the lowest and dominant harmonics is ₀ = π^−3∕2. For each L, one constructs the potential harmonic as being the linear combination which is maximally coupled to ₀. It is

(5.20)

where the summation runs over the magnetic numbers associated with L. It is coupled to ₀ through the transition potential

(5.21)

The word

potential comes from _L containing the harmonics that are present in the expansion of the potential itself. For power-law potentials, _L is independent of ξ. For general potentials, _L depends on ξ, but the method remains applicable [53].

In practice, one can first compute the bound state in the approximation of potential harmonics, with a fairly large L_max. Then the effect of non-potential harmonics of low L, say _[L], can be estimated by solving Sternheimer-type equations. If

(5.22)

at first order

(5.23)

where E₀ is frozen to its value at the approximation of potential harmonics. The second-order correction to the energy is

(5.24)

In practice, one often neglects the last term in Eq. (5.22), which only contributes to higher orders: then the corrections can be computed separately for each non-potential harmonic.