By Paul Bracken

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Thus, the field concept is at the forefront, playing a major role. Let us recall some important assumptions [1, 13–15] concerning the central problem of variational calculus: a) To find an extremal curve xi = xi (t) that satisfies δ Ldt = 0 requires that we restrict ourselves to a simply-connected domain. Though apparently technical, this point might entail a profound physical significance. 5772/53843 b) An extremal curve exists only in case that it can be embedded in a whole set of extremals, a so-called “Mayer field”.

E− 2 (λr )ℓ+1 L2nℓ+1 (λr )Yℓm (θ, ϕ) (2ℓ + n + 1) ! (4) with a scaling parameter λ, the Schrödinger equation of the AKP becomes a matrix equation: ←→ ← → 2 ←→ → ∂ E← 1 − λ ∆ (3) + ( 1 − γ ) λ − 2 Ψ = Id Ψ r λ ∂z2 (5) ←→ 2 ←→ ← → ← → ∂ 2 M Ψ ≡ −∆(3) + (1 − γ) 2 − ǫ Id Ψ = Ψ. λ ∂z (6) Dividing the whole equation by λ and packing E/λ2 into a parameter ε, one obtains This ǫ is to be fixed at some constant value. In principle any value will do, but for finite size of Hamiltonian matrix, the best choice is given [17] approximately 1 ε∗ ≃ − γ.

8) After fixing ε, the diagonalization of (6) is performed for 2/λi s and finally we obtain the energy eigenvalues by Ei = ελ2i . 3. Matrix diagonalization in Sturmian basis For the (tensored) harmonic wave function basis (THWFB) [11] we convert the Hamiltonian of AKP into the Hamiltonian of two of two-dimensional harmonic oscillators. 5772/55208 and the AKP Schrödinger equation becomes − 2 ( µ2 1 2 1 − γ ∂2 (2) (2) − 2 + ∆µ + ∆ν |Ψ = E |Ψ 2 2 ∂z2 +ν ) µ + ν2 . (11) Multiplying by µ2 + ν2 and swapping the Coulombic interaction term and the E term one obtains − 1 ∂2 1−γ (2) (2) + | E | µ2 + ν2 + ∆µ + ∆ν µ2 + ν2 |Ψ = 2 |Ψ .