A complex pole of the meromorphically continued resolvent or scattering matrix that represents a metastable or quasi‑bound state in a scattering problem; its real part gives the oscillation frequency and its imaginary part the decay rate.

Principle

Analytic continuation of the resolvent or S-matrix across the continuous spectrum produces isolated poles whose locations encode long‑time exponential decay and phase information for metastable modes.

Demonstration

For a compact obstacle in R^n or a short‑range potential, extend the outgoing resolvent across the positive real axis onto a nonphysical sheet; a pole close to the real axis corresponds to a long‑lived scattering mode that causes narrow peaks in the scattering cross section and an exponential tail in the time evolution.

Misapplication

Calling any peak in the measured scattering cross section a resonance without verifying the pole of a meromorphically continued operator, or treating a resonance pole as an actual L^2 eigenvalue, which ignores radiation and exponential decay.

Consequence

Identification of resonances yields quantitative lifetimes and phase shifts for metastable states, refines asymptotic expansions of scattering amplitudes, and enters trace formulas relating classical periodic orbits to spectral data.

Reversal

A true bound state (an L^2 eigenvalue) is the reversal: it lies on the real axis (or the physical sheet) and does not represent exponential decay into the exterior; virtual states or threshold phenomena are alternative nonresonant behaviors.

Boundary

Applies to linear scattering systems for which one can meromorphically continue the resolvent or S-matrix (e.g., short‑range potentials, obstacles, some black‑box settings); excludes settings lacking an analytic continuation or where only heuristic resonance notions (peaks) are used.

Semantic Tension

Tension exists between the analytic definition (poles of a continued operator) and the empirical notion of a resonance as a peak in response functions; closely related are alternative notions such as virtual states, embedded eigenvalues, and semiclassical resonances tied to classical trapped sets.

Synthesis

A scattering resonance is the complex spectral datum obtained from analytic continuation whose proximity to the real axis quantifies metastability: it unifies analytic continuation, asymptotic time behaviour, and observable scattering features into a single pole description.