  Using an operator-theoretic framework in a Hilbert-space setting, we perform
a detailed spectral analysis of the one-dimensional Laplacian in a bounded
interval, subject to specific non-self-adjoint connected boundary conditions
modelling a random jump from the boundary to a point inside the interval. In
accordance with previous works, we find that all the eigenvalues are real. As
the new results, we derive and analyse the adjoint operator, determine the
geometric and algebraic multiplicities of the eigenvalues, write down formulae
for the eigenfunctions together with the generalised eigenfunctions and study
their basis properties. It turns out that the latter heavily depend on whether
the distance of the interior point to the centre of the interval divided by the
length of the interval is rational or irrational. Finally, we find a closed
formula for the metric operator that provides a similarity transform of the
problem to a self-adjoint operator.
