Boundary Weyl Pseudospectra and Condition Numbers of Quasinormal Modes

Boundary Weyl Pseudospectra and Condition Numbers of Quasinormal Modes develops a structured pseudospectral diagnostic for black-hole and AdS quasinormal modes. Instead of perturbing the full bulk evolution operator, the paper studies sensitivity to perturbations of the boundary response. Given a Weyl, or Dirichlet-to-Neumann, function M(s) for the radial or bulk boundary value problem and a boundary law B(s), the quasinormal spectrum is encoded by the analytic boundary pencil F(s)=M(s)−B(s). The manuscript defines the boundary pseudospectrum through the small singular values of F(s) and derives the corresponding simple-pole condition-number formula for quasinormal-mode displacement under boundary-law perturbations. It shows that boundary exceptional points and boundary-driven ill-conditioning are detected by small values of the denominator ⟨ℓ,F′(s*)r⟩. The same denominator also appears in the residue of source-driven bulk response, so it controls, after multiplication by the appropriate adjoint-channel source overlap, the boundary-induced amplitude with which an infalling body or other localized source excites a ringdown mode. Using the adjoint-pair Krein resolvent formula, the paper relates the boundary pseudospectrum to the boundary-induced component of the bulk resolvent. This makes the boundary Weyl pseudospectrum a Schur-complement diagnostic for the part of quasinormal-mode sensitivity generated by boundary response. The construction is complementary to bulk-operator pseudospectra and to explicit Robin or double-trace spectral-flow analyses; it is not intended to replace either. A finite-worldtube Schwarzschild–AdS4 scalar example is included to illustrate the boundary-pseudospectral contours and to verify the first-order displacement law for a small boundary perturbation.