  Chaotic Hamiltonians are known to follow Random Matrix Theory (RMT) ensembles
in the apparent randomness of their spectra and wavefunction statistics.
Deviations form RMT also do occur, however, due to system-specific properties,
or as quantum signatures of classical chaos. Scarring, for instance, is the
enhancement of wavefunction intensity near classical periodic orbits, and it
can be characterized by a local density of states (or local spectrum) that
clearly deviates from RMT expectations, by exhibiting a peaked envelope, which
has been described semiclassically. Here, the system is connected to an
opening, the local density of states is introduced for the resulting
non-Hermitian chaotic Hamiltonian, and estimated a priori in terms of the
Green's function of the closed system and the open channels. The predictions
obtained are tested on quantum maps coupled both to a single-channel opening
and to a Fresnel-type continuous opening. The main outcome is that strong
coupling to the opening gradually suppresses the energy dependence of the local
density of states due to scarring, and restores RMT behavior.
