Mathematical Methods for Physics and Engineering - Matematica.NET

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS17.6 A useful generalisationSometimes we encounter inhomogeneous equations of a form slightly more generalthan (17.1), given byLy(x) − µρ(x)y(x) =f(x) (17.53)for some Hermitian operator L, with y subject to the appropriate boundaryconditions and µ a given (i.e. fixed) constant. To solve this equation we expandy(x) andf(x) in terms of the eigenfunctions y n (x) of the operator L, which satisfyLy n (x) =λ n ρ(x)y n (x).Working in terms of the normalised eigenfunctions ŷ n (x), we first expand f(x)as follows:∞∑∫ bf(x) = ŷ n (x) ŷn(z)f(z)ρ(z) ∗ dzUsing (17.29) this becomes=f(x) =n=0∫ ba∫ ba= ρ(x)ρ(z)ρ(x)a∞∑ŷ n (x)ŷn(z)f(z) ∗ dz. (17.54)n=0∞∑ŷ n (x)ŷn(z)f(z) ∗ dzn=0∞∑ŷ n (x)n=0∫ baŷ ∗ n(z)f(z) dz. (17.55)Next, we expand y(x)asy = ∑ ∞n=0 c nŷ n (x) and seek the coefficients c n . Substitutingthis and (17.55) into (17.53) we have∞∑∞∑∫ bρ(x) (λ n − µ)c n ŷ n (x) =ρ(x) ŷ n (x) ŷn(z)f(z) ∗ dz,n=0from which we find thatn=0n=0∫ ∞∑ ba ŷ∗ nn=0n=0(z)f(z) dzc n =.λ n − µHence the solution of (17.53) is given by∞∑∞∑∫ŷ n (x) by = c n ŷ n (x) =ŷ ∗λ n − µn(z)f(z) dz =From this we may identify the Green’s function∞∑ ŷ n (x)ŷ ∗G(x, z) =n(z)λ n − µ .an=0572a∫ ba∞∑n=0ŷ n (x)ŷn(z)∗ f(z) dz.λ n − µ