  Fractional (L\'{e}vy-type) operators are known to be spatially nonlocal. This
becomes an issue if confronted with a priori imposed exterior Dirichlet
boundary data. We address spectral properties of the prototype example of the
Cauchy operator $(-\Delta )^{1/2}$ in the interval $D=(-1,1) \subset R$, with a
focus on functional shapes of lowest eigenfunctions and their fall-off at the
boundaries of $D$. New high accuracy formulas are deduced for approximate
eigenfunctions. We analyze how their shape reproduction fidelity is correlated
with the evaluation finesse of the corresponding eigenvalues.
