Hello,

let's consider a compact and connected Riemannian manifold with the Schrödinger Operator $L=-\Delta +V:dom(H)\subset L^2(M)\rightarrow L^2(M)$ whereas $dom(L):=\lbrace f\in C^{\infty}(M,\mathbb{R}) \vert f_{\vert \partial M}=0 \rbrace$ and $V\in L^2(M)$ bounded.

The spectrum of the Friedrichs extension of $L$ consists of a discrete set of Eigenvalues $(\lambda_i)_{i=1}^{\infty}$ with corresponding eigenfunctions $\phi_i$, which form an $L^2$-orthonormal basis.

I want to know, why the eigenvalues $\phi_i$ vanish on the boundary , i.e. $\phi_i (x)=0 \forall x\in\partial M$. I hope, you can explain it to me.

Regards.