I want to solve this eq. (k(x)*g(x)')'=p*g(x)

[simplex]

I have the following equation:

(k(x)*g(x)')'=p*g(x)

where
k(x) = k(x+T) -- k(x) is a known periodical function of period T, k(x) real, x real, T real.
p = some constant that have to be determined.
g(x) = an unknown function.

Question: Is there a method that can tell from the beginning that all the values of p should be real?

If p is always real I will try to solve the equation if p could also be complex the equation will not be of so much interest for me.

 

[simplex]

Can somebody prove that the operator:

H = d/dx(k(x)*d/dx) corresponding to the above mentioned equation, Hg(x)=p*g(x), is hermitian?

I understand that if you prove this then all the values of p must be real.