Let $A$ be a possibly unbounded skew-adjoint operator on the Hilbert space $X$ with compact resolvent. Let $C$ be a bounded operator from $\Dscr(A)$ to another Hilbert space $Y$. We consider the system governed by the state equation $\dot z(t)=Az(t)$ with the output $y(t)=Cz(t)$. We characterize the exact observability of this system only in terms of $C$ and of the spectral elements of the operator $A$. The starting point in the proof of this result is a Hautus type test, recently obtained in Miller \cite{Miller}. We then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain. The Schrödinger equation, the Bernoulli-Euler plate equation and the wave equation in a square are considered. For the plate and Schrödinger equations, the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary. This is done by combining our spectral observability test to a theorem of Beurling on non harmonic Fourier series and to a new number theoretic result on shifted squares.
