(a) Show that if L is a self-adjoint operator, then so is eL.(b... | Filo

SOLVED:(a) Suppose that the self-adjoint operator 𝒮 in Problem 3(b) is defined on a space of functions satisfying the conditions a1 X(a) a2 X^'(a)=0, b1 X(b) b2 X^'(b)=0 where a1 and a2 are not both zero and where the same is true of b1 and b2. Use Lagrange's identity, obtained in that problem, to show that (X, ℒ[Y])=(ℒ[X], Y) where these inner products are on the interval a<x<b and with weight function unity. (b) Let λm and λn denote distinct eigenvalues of a regular Sturm-Liouville problem,