# Applying boundary conditions for differential operator to show operator is hermitian.

I'm trying to determine when the differential operator $L[f] = - \frac{d^2f}{dx^2} = - f''(x)$ is Hermitian, for different types of boundary conditions on $f$; however, I'm unclear how to interpret the boundary conditions in this context.

I'm not sure if this is because the question is loosely phrased and requires interpretation, or if it is just my own ignorance, so here is the original question:

Verify that $L[f] = -\frac{d^2f}{dx^2}$ is Hermitian on $[a,b]$ if $f$ satisfies (a) Dirichlet, (b) Neumann, [c] Periodic or (d) Robin boundary conditions.

I was able to show that

\begin{eqnarray*} \left<\, L[y_1], \,y_2 \right> = \left<\, y_1, \, L[y_2]\right> - \underbrace{\vphantom{\bigg|}y_1\mkern{-0.16em}' \,y_2\big|_a^b + y_1 \,y_2\mkern{-0.16em}'\big|_a^b}_{\text{boundary-dependent}} \end{eqnarray*}

however, I'm not clear what it would mean for $f$ ... satisfies (a) Dirichlet ... boundary conditions'' at this point in my work? Does that mean:

a) $y_1(a) = c_1$, $y_1(b) = c_2$ and $y_2(a) = k_1$, $y_2(b) = k_2$

or

b) $y_1(a) = y_2(a) = c_1$ and $y_1(b) = y_2(b) = c_2$

• (a) $f = 0$ on the boundaries (b) $f' = 0$ on the boundaries (c) $f' - \alpha f = 0$ for $\alpha \neq 0$ (d) $f(a) = f(b)$ and $f'(a) = f'(b)$, You should be able to easily verify whether or not the boundary term indeed vanishes for each of these cases. – Gregory Jun 2 '17 at 0:51