Domain of adjoint operator of negative Laplacian on an interval

Question

Consider the negative Laplacian $-\Delta:=-\frac{d^2}{dx^2}$ on $L^2(0,1)$ with domain $D(-\Delta)=C_0^\infty(0,1)$, where $C_0^\infty(0,1)$ is the set of all smooth function with compact support on $(0,1)$: $$ -\Delta:L^2(0,1)\supset C_0^\infty(0,1)\to L^2(0,1). $$ I read that the adjoint operator $-\Delta^*$ has the domain $D(-\Delta^*)=H^2(0,1)$, but I never saw a thorough proof. Does anybody know a source where this case is discussed in detail?