I have been given an exam question asking to show that the spectral representation of the Green's Function for a given Hermitian Operator admits solutions of a given form. I have an answer, but the catch is my university doesn't give ANY solutions to past exams, so I'm not sure whether to back my method or not. There is a another method in my notes, but I just wondered whether or not this solution is legitimate.
My solution goes as follows...
$$G(x,t)=\sum_n \frac1\lambda_n \phi_n(x)\phi_n(t)$$
Consider the differential equation:
$$L(x)y(x)=f(x)$$
$L(x)$ admits a complete set of orthogonal eigenfunctions, so we may write:
$$y(x)=\sum_na_n\phi_n(x),f(x)=\sum_nb_n\phi_n(x)$$
Applying $L(x)$ to $y(x)$ yields:
$$L(x)y(x)=\sum_na_nL(x)\phi(x)=\sum_na_n\lambda_n\phi_n(x)=f(x)=\sum_nb_n\phi_n(x)$$
Leading to:
$$a_n\lambda_n=b_n$$
Multiplying $G(x,t)$ by $f(t)$ gives:
$$G(x,t)f(t)=\sum_n\frac1\lambda_n\phi_n(x)\phi_n(t)b_n\phi_n(t)$$
Using the relation between $a_n$ and $b_n$:
$$=\sum_na_n\phi_n(x)\phi_n(t)\phi_n(t)$$
Integrating:
$$\int G(x,t)f(t)dt=\int \sum_na_n\phi_n(x)\phi_n(t)\phi_n(t)dt$$ $$=\sum_na_n\phi_n\int \phi_n(t)\phi_n(t)dt$$
Noticing that this integral is just the inner product of a real function with itself, we conclude that, if the eigenfunctions are normalized, the inner product is just 1. This leads to the required result:
$$\int G(x,t)f(t)dt=\sum_na_n\phi_n(x)=y(x)$$
I'm not sure whether or not there is any spurious reasoning anywhere, so I'd appreciate a more experienced/confident mathematician's approval!
Thanks in advance.