3.7

Suppose f and g are eigenfunctions of Q, with eigenvalue q. Show any linear combination of f and g are eigenfunctions of Q with eigenvalue q.

|Qf> = q|f>
|Qg> = w|g>

A and B are (possibly complex) constants.

|Q(A*f+B*g)> = A|Qf> + B|Qg> = Aq|f> + Bq|g>
	= q(A|f> + B|g>).

QED

Check that f(x)=exp(x) and g(x)=exp(-x) are eigenfunctions of the operator d^2/dx^2, with the same eigenvalue. construct two linear combiations of f and g that are orthogonal eigenfunctions on the interval [-1,1]


if Q = d^2/dx^2 then we have the differential equation

f'' = qf

(e^x)'' = e^x, so this is an eigenfunction with q=1.

(e^-x)'' = e^-x, so this is an eigenfunction with q=1.

e^x + e^-x is also an eigenfunction of q, per earlier proof, and so is e^x - e^-x. These functions are just 2*sinh and 2*cosh, which are orthogonal functions because sinh is odd and cosh is even.