Mathematical Methods for Physics and Engineering - Matematica.NET

17.7 EXERCISESWe note that if µ = λ n ,i.e.ifµ equals one of the eigenvalues of L, thenG(x, z)becomes infinite and this method runs into difficulty. No solution then existsunless the RHS of (17.53) satisfies the relation∫ baŷ ∗ n(x)f(x) dx =0.If the spectrum of eigenvalues of the operator L is anywhere continuous,the orthonormality and closure relationships of the normalised eigenfunctionsbecome∫ ba∫ ∞0ŷ ∗ n(x)ŷ m (x)ρ(x) dx = δ(n − m),ŷ ∗ n(z)ŷ n (x)ρ(x) dn = δ(x − z).Repeating the above analysis we then find that the Green’s function is given byG(x, z) =∫ ∞0ŷ n (x)ŷ ∗ n(z)λ n − µdn.17.7 Exercises17.1 By considering 〈h|h〉, whereh = f + λg with λ real, prove that, for two functionsf and g,〈f|f〉〈g|g〉 ≥ 1 [〈f|g〉 + 4 〈g|f〉]2 .The function y(x) is real and positive for all x. Its Fourier cosine transform ỹ c (k)is defined by∫ ∞ỹ c (k) = y(x)cos(kx) dx,−∞and it is given that ỹ c (0) = 1. Prove thatỹ c (2k) ≥ 2[ỹ c (k)] 2 − 1.17.2 Write the homogeneous Sturm-Liouville eigenvalue equation for which y(a) =y(b) =0asL(y; λ) ≡ (py ′ ) ′ + qy + λρy =0,where p(x),q(x) andρ(x) are continuously differentiable functions. Show that ifz(x) andF(x) satisfyL(z; λ) =F(x), with z(a) =z(b) =0,then∫ bay(x)F(x) dx =0.Demonstrate the validity of this general result by direct calculation for thespecific case in which p(x) =ρ(x) =1,q(x) =0,a = −1, b =1andz(x) =1− x 2 .17.3 Consider the real eigenfunctions y n (x) of a Sturm–Liouville equation,(py ′ ) ′ + qy + λρy =0, a ≤ x ≤ b,in which p(x), q(x) andρ(x) are continuously differentiable real functions andp(x) does not change sign in a ≤ x ≤ b. Takep(x) as positive throughout the573