While revisiting the notion of a cyclic vector for a self-adjoint operator (see, e.g., Schmuedgen, Unbounded Self-adjoint Operators on Hilbert space, Section 5.4), I got stuck when I tried to figure out a couple of concrete examples on 'everyday life' operators.
Consider the Dirichlet Laplacian on an interval, say, the operator $T=-\frac{d^2}{dx^2}$ acting on the Hilbert space $L^2(0,1)$ with domain (of self-adjointness) $D(T)=\{f\in H^2(0,1)\,|\,f(0)=f(1)=0\}$.
QUESTION A. Does $T$ admit a cyclic vector? In other words (Schmuedgen, Definition 5.1), is there a function $\varphi\in D(T)$ (as well as $\varphi\in D(T^n)$ for any positive integer $n$) such that the span of $\varphi,T\varphi,T^2\varphi,\dots$ is actually dense in $L^2(0,1)$? I understand that this is tantamount as asking whether $T$ has simple spectrum (Schmuedgen, Definition 5.1).
QUESTION B. Does $T$ admit a cyclic vector $\varphi$ that belongs to $C^\infty[0,1]$ and vanishes, together with all its derivatives, at $x=0$ and $x=1$ ?
(My motivation: it is easy to decide the existence of cyclic vectors for operators in multiplication form, I was trying to practice with differential operators instead.)