Reflectionless Transmission Through Dielectrics And Scattering Potentials

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Reflectionless Transmission Through Dielectrics And Scattering Potentials
Irvin W Kay
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6) i-zzv^M ^^nV^M " z:pn%^"pM n •- -* n, v •- -* By Lemma 2 the a and p are positive^ therefore, (3. 6) is positive. This com- pletes the proof of Theorem 2, Finally, let us examine the asymptotic behavior of V(x), We state Theorem 3 . V(x) approaches zero exponentially as x approaches either + 00 or -oo» Proof. From the proof of Theorem 2 we have for the form of V(x) : - 10 - V(x) = - 2 .
n •- -* n, v "- -"I (3. 7) n *- -• rijV •- -J -1 > • Now suppose p is the largest of the constants p • Ihe
...n the term in the denomina- tor of (3»7) of largest exponential grovrth as x approaches +oo is that for which n = V = |i, that is, for which the exjjonent is 2p . This term is obviously missing from the numerator of (3e7)j thus, the denominator grows faster than the ntimera- tor, and V(x) approaches zero as x approaches +co» As x approaches -oo, V(x) obviously approaches zero.
li» Example Consider the case of one constant A_ and one constant x, • The sytem (2o2) will consist of a single equation: (U.


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