| Description: |
Evolution systems driven by non-selfadjoint operators present certain genuinely dissipative dynamical features, absent in conservative systems. In a gravitational setting, the choice of a hyperboloidal foliation for the description of wave dynamics on a background spacetime with outgoing boundary conditions (e.g. black holes), casts the (linear) dynamics in terms of an evolution problem whose infinitesimal time generator is a non-selfadjoint operator. A general framework for non-conservative systems is given in terms of i) the spectral theory of non-selfadjoint operators and ii) the non-modal analysis associated with non-normal dynamics. They provide, respectively, complementary spectral and time-domain descriptions of non-conservative dynamics, sharing the key feature of being defined for normalisable states in a Hilbert (Banach) space. We focus here on two genuinely non-selfadjoint problems, illustrating this spectral/time-domain complementarity, namely: a) the construction of asymptotic resonant expansions and b) the assessment of dynamical linear transients. Regarding a), the so-called Keldysh expansion of the resolvent permits to express time-evolved fields as quasinormal mode expansions, with quasinormal frequencies characterised as eigenvalues of the non-selfadjoint operator. As a result, we provide general closed expressions for the spectral decomposition of the field that generalise (in an asymptotic sense) the standard (spectral-theorem) projection scheme in self-adjoint problems. Regarding b), and making use of Sobolev norms weighting high-derivatives to assess the regularity/small-scale structure of solutions, we demonstrate the presence of an initial growth in the Sobolev-norm of evolving (linear) fields, indicating an initial transitory enhancement of the “small scale/regularity”. |