The equation under consideration is

(d²φ/dx²) = −k²(x)φ(x)

where k(x) is a positive and symmetric about x=0 but is a declining function of |x|. This equation can be termed a generalized Helmholtz equation.

Now consider the points of relative maxima and minima of φ², labeled as x_i and y_i, respectively. These are points such that dφ²/dx=2φ(dφ/dx) equals zero; (dφ/dx)=0 for the maxima and φ=0 for the minima. Note that because of the equation, when φ=0 then d(dφ/dx)/dx=0 and hence (dφ/dx) is at a maximum or minimum.

Let the labeling be such that x_i > y_i.

A Relationship Between the
Maxima of φ² and (dφ/dx)²

Multiply the equation by (dφ/dx) to get

(dφ/dx)(d²φ/dx²) = −k²(x)φ(x)(dφ/dx)
which is the same as
½(d((dφ/dx)²)/dx) = −½k²(x)(d(φ²)/dx)

Now integrate the above equation from x=y_i to x=x_i. The result may be represented as

½[((dφ/dx)²]_yi^x_i = −½∫_yi^x_ik²(x)(dφ²/dx)dx

At x_i (dφ/dx) is equal to 0. By the Generalized Mean Value Theorem the RHS of the above equation may be expressed as

= −½∫_yi^x_ik²(x)(dφ²/dx)dx = −½k²(z_i)∫_yi^x_i(dφ²/dx)dx
which reduces to
−½k²(z_i)[φ²]_yi^x_i

where z_i is some point between y_i and x_i.

But φ² is equal to zero at y_i and (dφ²/dx) is equal to zero at x_i.

Therefore the previous equation reduces to

−½(dφ/dx)²_yi = −½k²(z_i)φ²(x_i)
or, equivalently
φ²(x_i) = (dφ/dx)²_yi/k²(z_i)
where y_i<z_i<x_i

As formulated φ²(x_i) is a local relative maximum of φ² and (dφ/dx)²_yi is the next lower local relative maximum of (dφ/dx)². The same relationship would prevail between a local relative maximum of φ² and the next lower local relative maximum of (dφ/dx)². Or, between a local relative maximum of (dφ/dx)² and the next lower and next higher local relative maximum of φ².

This gives

(dφ/dx)²_yi+1 = k²(w_i)φ²(x_i)
where x_i<w_i<y_i+1

This means that

φ²(x_i+1) = [k(z_i)/k(w_i)]φ²(x_i)
where
z_i<x_i<w_i<y_i+1

This provides a chain of ratios that links φ²(x_i) to some φ²(x₀). If z_i+1≅w_i the formula would reduce to

φ²(x_i) = φ²(x₀)/k(w_i)

Instead

φ²(x_i) = φ²(x₀)Γ_i/k(w_i)
where
Γ_i = [k(z₁)/k(w₀)]·…[k(z_i)/k(w_i-1)]

The factor Γ_i is the product of i terms each of which is less than one.

Some Approximations

If k(x)² were constant over some interval of x about x₀ then over that interval

φ(x) = A·cos(k(x−x₀))

where A is a constant. The wavelength L is given by

L = 2π/k

The difference between y_i and x_i is then ½L=π/k, and

z_i ≅ x_i − ¼L
and
w_i ≅ x_i + ¼L
z_i+1 ≅ x_i+1 − ¼L

A more convnient representation is

z_j = y_j + ½L_j
and
w_j-1 = y_j − ½L_j

where L_j=2π/k(y_j).

The simplified formula previously derived for φ²(x_i) can be expressed as

φ²(x_i) = φ²(x₀)Γ_i/k(x_i+½L_i)

where

Γ_i = Π [k(y_j+½L_j)/k(y_j−½L_j)]

For some cases φ² is at a local maximum at x=0; for other cases it is at a local minimum there.

The average value of φ²(x) in the interval about x_i is equal to ½φ²(x_i).

Asymptotics

As E → ∞ so k(x)→∞ for all x. Thus L_j → 0 and hence Γ_j → 1 for all j. Likewise (x_i+½L_i)→x_i.

Roughly then, the average value of φ²(x) asymptotically is inversely proportional to k(x).