Error budgets from pseudospectra for black hole ringdown: a Schwarzschild baseline

Abstract We present a compact and reproducible matrix level pseudospectral audit of Schwarzschild perturbations designed to quantify sensitivity beyond eigenvalue convergence for semi discrete evolution operators. From the tortoise coordinate master equation we construct a first order formulation and a finite difference generator with outgoing boundary enforcement. We compute the smallest singular value field $$\sigma _{\min }(A_N-sI)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:mi>s</mml:mi> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> on a prescribed complex s plane window and extract nested $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> pseudospectrum bands for the Regge Wheeler axial gravitational sector $$(\ell =2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and a massless scalar sector $$(\ell =1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The reported bands, pseudospectral abscissae, and migration radii refer specifically to the stated state variables, Euclidean matrix norm, finite interval, resolution, and penalty implementation; they are not asserted to be invariants of the continuum Schwarzschild quasinormal mode operator. The standard bands provide unstructured matrix perturbation bounds and are not interpreted as direct physical uncertainty intervals. We also evolve a compact initial pulse, extract a detector waveform, report the Euclidean state norm evolution for that pulse, and resolve time frequency content with a short time Fourier transform spectrogram. The results exhibit matrix resolvent sensitivity and time localized transient structure not summarized by eigenvalues alone, together with sector dependent differences consistent with effective barrier structure.