Sturm-Liouville operator with Dirichlet BC

Question

I am trying to understand why Sturm-Liouville operator $$L(f)(x)=f''(x)-p(x)f(x)$$ with Dirichlet boundary conditions on $[a,b]$ is unbounded.

$f$ is twice continuously differentiable, $p(x)>0$ is continuous. Where does the trouble come from?

Answer 1

Let $f_n = \sin\left(n\pi\frac{x-a}{b-a}\right)$. Then $$ Lf_n = -\left(\frac{n^2\pi^2}{(b-a)^2}+p\right)f_n \\ $$ If $P$ is a uniform bound for $p$ on $[a,b]$, and if $n$ is large enough, $$ |Lf_n| \ge \left(\frac{n^2\pi^2}{(b-a)^2}-P\right)|f_n| \\ \|Lf_n\| \ge \left(\frac{n^2\pi^2}{(b-a)^2}-P\right)\|f_n\| $$