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AMATH 581 ( c○J. N. Kutz) 21n(x)/n 010.80.60.40.20−3 −2 −1 0 1 2 3xFigure 7: Plot of the spatial function n(x).Eigenvalues and Eigenfunctions: The Infinite DomainBoundary value problems often arise as eigenvalue systems for which the eigenvalueand eigenfunction must both be determined. As an example of such aproblem, we consider the second order differential equation on the infinite lined 2 ψ ndx 2 +[n(x) − β n] ψ n =0 (1.3.6)with the boundary conditions ψ n (x) → 0asx →±∞. For this example, weconsider the spatial function n(x) whichisgivenby{1 −|x|20 ≤|x| ≤1n(x) =n 00 |x| > 1(1.3.7)with n 0 being an arbitrary constant. Figure 7 shows the spatial dependence ofn(x). The parameter β n in this problem is the eigenvalue. For each eigenvalue,we can calculate a normalized eigenfunction ψ n . The standard normalizationrequires ∫ ∞−∞ |ψ n| 2 dx =1.Although the boundary conditions are imposed as x →±∞,computationallywe require a finite domain. We thus define our computational domain to bex ∈ [−L, L] whereL ≫ 1. Since n(x) =0for|x| > 1, the governing equationreduces tod 2 ψ ndx 2 − β nψ n =0 |x| > 1 (1.3.8)which has the general solutionψ n = c 1 exp( √ β n x)+c 2 exp(− √ β n x) (1.3.9)