A complex pole of the meromorphically continued resolvent or scattering matrix (or the analytic continuation of the Green's function) that represents quasi-bound or long-lived oscillatory modes associated with an obstacle, potential, or boundary; resonances generalize eigenvalues to open systems and encode decay rates and oscillation frequencies.

Principle

Resonances occur where the analytically continued inverse (resolvent) fails to be analytic; their real parts indicate oscillation frequency and imaginary parts (typically negative) indicate exponential decay rates of the associated quasi-modes.

Demonstration

In exterior scattering by a compact obstacle in R^n, the scattering matrix continued into the complex plane has poles whose positions near the real axis correspond to long-lived scattering states; in physics these are seen as peaks in cross-sections and as slowly decaying wave packets.

Misapplication

Calling any peak in a scattering amplitude a resonance without verifying the existence of a nearby pole, or confusing a true resonance (pole) with a bound state (real eigenvalue) or a threshold effect.

Consequence

Resonances determine the late-time asymptotics of wave propagation and contribute poles in spectral expansions; they control decay rates, resonance expansions, and capture metastable phenomena in open systems.

Reversal

The absence of scattering resonances near the real axis implies faster dispersion and no long-lived quasi-modes; in closed (compact) systems the analogue are genuine eigenvalues on the real axis rather than complex resonances.

Boundary

Definition presumes a framework allowing analytic continuation (e.g. meromorphic continuation of the resolvent via complex scaling or analytic Fredholm theory); in purely discrete or non-analytic settings the term may reduce to eigenvalue theory.

Semantic Tension

Competes semantically with 'eigenvalue' and with heuristic uses of 'resonance' in experimental spectra; mathematically resonances are analytically defined poles, not merely observed peaks or approximate numerical artifacts.

Synthesis

Scattering resonances are complex spectral poles of analytically continued scattering operators that encode frequency and decay of metastable states in open systems; they bridge eigenvalue notions for closed systems and observable long-time scattering behavior.