We apply the complex scaling method to black-hole perturbations in four-dimensional Schwarzschild--de~Sitter (dS) spacetimes. The method converts the outgoing-wave boundary-value problem into a non-Hermitian spectral problem and enables quasinormal-mode poles and the rotated continuum to be treated in a common framework. We focus in particular on the continuum level density, which characterizes the continuum response beyond isolated quasinormal-mode frequencies. Using Regge--Wheeler-type perturbation equations for scalar, electromagnetic, and gravitational fields, we investigate how a nonzero cosmological constant modifies the pole and continuum sectors. We also discuss a possible extension to string-inspired coupled-channel systems, and illustrate that higher-dimensional dS black holes can be treated within the same framework, at least in tensor- and vector-type sectors. Our results indicate that complex scaling offers a useful spectral framework for analyzing both quasinormal modes and continuum response in black-hole physics.
We study the quasinormal-mode and quasibound-state spectra of a massless scalar field in weakly magnetized Ernst--Schwarzschild spacetime. In the separated
O(B02) regime, the Klein--Gordon equation reduces to a radial confluent-Heun equation. We construct an endpoint-selected spectral Wronskian directly from the canonical confluent-Heun recurrence and its regular-to-irregular connection limit. Quasinormal modes and quasibound states are then obtained as zeros of the same analytic spectral function on the radiative and bound sheets, respectively, providing a unified treatment of the two spectral sectors within the separated weak-field problem. The resulting QNM and QBS frequencies agree with independent weak-field Ernst and massive-Schwarzschild benchmarks available in the literature. We further show, for representative cases, that the frequencies selected by the local Heun-series termination relation examined here do not satisfy the endpoint-selected quasibound-state condition. The construction therefore provides a direct horizon-to-infinity spectral condition for both radiative and bound states within the weak-field Ernst problem.
Based on the approach of biorthogonal basis, we carry out the quasinormal modes (QNMs) expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the closed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H † . The time-independent perturbation theory for the complex frequencies can be also developed.
We investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters,
(ε,d,σ0), a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. Notably, the EL exhibits an anisotropic spectral response: parameters migrating along the EL direction leaves the coalesced QNM spectra nearly unchanged, while moving parameters away from the EL induces the characteristic
ε1/2 scaling, highlighting the directional nature of spectral instability in exceptional structures. We find that the vorticity
ν=±1/2 and the Berry phase
γ=π for loops encircling the EL, while
ν=0 and
γ=0 for those do not encircle the EL. In the neighborhood of an eigenvalue, through matrix perturbation theory, we prove that the
ε-pseudospectrum contour size scales as
ε1/q at an EP , where
q is the order of the largest Jordan block of the Hamiltonian-like operator associated with that eigenvalue, contrasting with the linear
ε scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.
We study quantum mechanical wavefunctions near highly curved spaces, i.e., black holes. By utilizing the formalism developed by DeWitt, we derive the Schrödinger equations in the vicinity of the Schwarzschild and the Reissner-Nordström black hole geometries. The quantum electron cloud for the "black hydrogen atom" - an electron trapped by black holes - is particularly studied. We solve the equations and find that black holes generally attract the wavefunctions, localizing them near the horizon where the electrons are most likely to be trapped. These results imply that not only classical objects but also the quantum material and even the chemical properties of the atoms are affected by strong gravity. We also discuss black hydrogen molecules composed of multi-centered Majumdar-Papapetrou black holes.
It is shown that the internal stationary state of the Schwarzschild black hole can be represented by a maximally entangled two-mode squeezed state of collapsing matter and infalling Hawking radiation. The final boundary condition at the singularity is then described by the random unitary transformation acting on the collapsing matter field. The outgoing Hawking radiation is obtained by the final-state projection on the total wave function, which looks like a quantum teleportation process without the classical information transmitted. The black hole evaporation process as seen by the observer outside the black hole is now a unitary process but nonlocal physics is required to transmit the information outside the black hole. It is also shown that the final-state projection by the evaporation process is strongly affected by the quantum state outside the event horizon, which clearly violates the locality principle.