Page 19 - Winter 2010

ble to gain significant understanding by seeing how to calcu- late a localized eigenfunction (and its eigenvalue) in an actu- al one-dimensional disordered system. While it may not be necessary except for a deeper understanding, it will be worth- while to review the mathematics which underlies the method for finding such an eigenfunction. Insight into Anderson localization can be gained even if the mathematical notions simply establish terminology. In any case, we will “bite the bullet” and delve into mathematics; the useful concepts are as follows:
1. The one-dimensional wave equation is a linear differ- ential equation involving second order derivatives. The dif- ferential equation will involve a parameter such that when boundary conditions are added (from the array of scatterers), then solutions may be found only for certain values of the parameter; this parameter becomes the eigenvalue, and the solution for that value is the eigenfunction ψ (x). For such a problem there are many math equations comprising what is
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pendent functions”—two functions f1 (x) and f2 (x) are lin-
early independent if and only if it is impossible to find two
nonzero constants C1 and C2 such that C1f1 (x) + C2f2 (x) = 0
for all x. Despite the mathematical formality of the definition,
such functions are common in acoustics; examples of pairs of
linearly independent functions are sin (kx) and cos (kx), exp
(ikx) and exp (−ikx), and exp (αx) and exp (−αx). It should be
pendent solutions, and any solution may be expressed as a linear combination of the two.
4. If one linearly independent function exponentially decays asymptotically, then the second linearly independent function will exponentially grow asymptotically. The impor- tant notion here is that for sufficiently extreme values of some variable, one of the two linearly independent solutions will be much larger than the other.
5. If the value of an eigenfunction and its first derivative are known at just one point [ψ (x0) and dψ/dx (x0) for some x0], then that is sufficient information to determine the eigenfunction ψ (x) at all points.
6. Furstenberg’s statement about exponential localization (with probability notions in an infinite system) will be assumed, so that we have some idea of the properties of the eigenfunction. It should be noted that the exponential local- ization does not mean that the eigenfunction is given asymp- totically by A exp (−α |x|), but rather that its amplitude is bounded by that function, for some value of A.
We now have enough mathematical concepts (or at least terminology) to develop an understanding of Anderson localization in one dimension. We begin by supposing that we know the exact value of one eigenvalue as well as the exact value and first derivative of its eigenfunction at one point. Then by statement 5 above, we can find the entire eigenfunc- tion. This is illustrated in Fig. 4a—the point where the value and first derivative of the eigenfunction are known is in the center. The math theorem means that the entire wave function can be found by proceeding in the positive and negative x-
known as Strum-Liouville theory.
2. It will be helpful to use the concept of “linearly inde-
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3. For the wave equation there exists two linearly inde-
noted that the last pair may represent evanescent waves.
Wave Propagation in Arrays 15