In their paper [1] Evans & Eckardt start with the ODE

                ^∂²Φ/_∂r² + (²/_r+ω_r) ^∂Φ/_∂r + (2rω_r + r² ^∂ω_r/_∂r) ^Φ/_r² = − ^ρ/_εo                                 (1)

which is a linear ODE with r-dependant coefficients. They assert:

Eq.(1) can be reduced straightforwardly to the basic structure of the damped oscillator equation, which was discovered in the eighteenth century {11}

                ^d²x/_dr² + 2β ^dx/_dr + κ_o² x = A cos (κr)                                                                 (2)

In Eq.(2) β takes the role of the friction coefficient and κ_o is a Hooke law type wave number. . . .

We remark that apparently Evans has given up his method of attaining resonance by a dubious complex "Euler transform" (see [2]) and instead of this failed method he now tries to get resonance by pure "definition". He continues:

Eq.(1) reduces to Eq.(2) when

                ω_r = 2 (β − ¹/_r)                                                                                                 (3)

                κ_o² = ⁴/_r (β − ¹/_r)                                                                                                 (4)

This is incorrect: The damped oscillator equation has constant coefficients while the coefficients of Evans' ODE (2) are given by Eqs.(3-4) and are evidently non-constant if we assume β≠¹/_r . However, already the case β=¹/_r shows the essential difference between the ODEs (1) and (2): Due to ω_r=0 and κ_o=0 we obtain from Eq.(1)

                ^∂²Φ/_∂r² + ²/_r ^∂Φ/_∂r + 0 ^Φ/_r² = − ^ρ/_εo                                                                 (1')

with the non-oscillatory eigensolutions

                Φ₁(r) = 1         and         Φ₂(r) = ¹/_r ,                                                                 (1")

while the damped oscillator equation (2) (with constant coefficient β) takes the form

                ^d²x/_dr² + 2β ^dx/_dr = A cos (κr)                                                                                 (2')

with the eigensolutions

                x₁(r) = 1         and         x₂(r) = e^−2βr                                                                 (2")

The comparison of the eigenfunctions (1") and (2") shows that the ODEs (1) and (2) disagree essentially.

In addition, if one takes the cofficients of ODE (2) at a fixed point r=r_o then the eigenfunctions of Eq.(2) depend on the value r_o. More, for β<0 or for β>0 and r<1/β the ODE (2) has no oscillatory eigenfunctions and therefore cannot yield resonance. Therefore it turns out that:

The resonance behavior of Eq.(1) cannot be discussed by means of the ODE (2), constant coefficients β and κ_o assumed.