Eigenfunctions and operators.?

Question

For each of the following functions determine whether the function is an eigenfunction of the d/dx and d^2/dx^2 operators, and if so determine the eigenvalues:
a) e^(-ax^2)
b) cos(bx)
c) e^(ikx) where i= sqrt (-1)

why wouldnt for a) d/dx be an eigenfunction? d/dx would equal -2axe^(-ax^2)......why wouldn't that be considered a multiple of e^(-ax^2)?

also for part c...isn't the d/dx equal to ike^ikx?, not -ike^ikx?

Answer 1

Let EV mean "eigenvalue"

a) For d/dx, no EV
For d^2/dx^2, no EV

b) For d/dx, no EV
For d^2/dx^2, EV = - b^2

c) For d/dx, EV = - i k
For d^2/dx^2, EV = - k^2

Let f(x) be a function, and let O(f) be an operator. If

O(f(x)) = k f(x)

Then f(x) is the eigenfunction of the operator, and k is the eigenvalue.