﻿ The relative extrema of the solution to a generalized Helmholtz equation
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

The Relative Extrema of the Solution
to a Generalized Helmholtz Equation

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 xi and yi, 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 xi > yi. ## 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=yi to x=xi. The result may be represented as

#### ½[((dφ/dx)²]yixi = −½∫yixik²(x)(dφ²/dx)dx

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

#### = −½∫yixik²(x)(dφ²/dx)dx = −½k²(zi)∫yixi(dφ²/dx)dx which reduces to −½k²(zi)[φ²]yixi

where zi is some point between yi and xi.

But φ² is equal to zero at yi and (dφ²/dx) is equal to zero at xi.

Therefore the previous equation reduces to

#### −½(dφ/dx)²yi = −½k²(zi)φ²(xi) or, equivalently φ²(xi) = (dφ/dx)²yi/k²(zi) where yi<zi<xi

As formulated φ²(xi) is a local relative maximum of φ² and (dφ/dx)²yi 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

This means that

#### φ²(xi+1) = [k(zi)/k(wi)]φ²(xi) where zi<xi<wi<yi+1 This provides a chain of ratios that links φ²(xi) to some φ²(x0). If zi+1≅wi the formula would reduce to

#### φ²(xi) = φ²(x0)Γi/k(wi) where Γi = [k(z1)/k(w0)]·…[k(zi)/k(wi-1)]

The factor Γi 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 yi and xi is then ½L=π/k, and

#### zi ≅ xi − ¼L and wi ≅ xi + ¼L zi+1 ≅ xi+1 − ¼L

A more convnient representation is

#### zj = yj + ½Ljand wj-1 = yj − ½Lj

where Lj=2π/k(yj).

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

where

#### Γi = Π [k(yj+½Lj)/k(yj−½Lj)]

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 xi is equal to ½φ²(xi).

## Asymptotics

As E → ∞ so k(x)→∞ for all x. Thus Lj → 0 and hence Γj → 1 for all j. Likewise (xi+½Li)→xi.