Draw a potential V(x). The scattering resonances of −u″ + Vu = λ²u — the poles of the continued resolvent — are found from the Jost function and shown in the complex λ‑plane. The third panel is the wave trace those resonances predict through the Poisson formula.
Click and drag to draw. Supported on −4 ≤ x ≤ 4; zero outside.
Resonances come in pairs λ, −λ̄, so the picture is symmetric about the imaginary axis. Closer to the real axis means longer lived.
Tr(UV(t) − U0(t)) = ½ ∑ e−iλjt − ½, summed over the resonances found at left. Every amplitude is ±½.
One of each mirror pair listed.
| # | Re λ | Im λ | lifetime 1/|Im λ| | residual | kind |
|---|
This runs the wave-trace fitting method end to end, entirely in this page: discretize −u″+Vu on a truncated interval, diagonalize, assemble the windowed trace, then recover nodes from the samples by certified deflation (c frozen at the nearest half-integer once it is certified, so the tail cannot corrupt nodes already found). No knowledge of the Jost function is used here — it is a genuinely independent check of the panels above.
How it works. Outside the support the solutions are exactly
e±iλx, so a solution outgoing at −∞ is propagated across
the potential with the exact 2×2 propagator of a piecewise constant potential, and
λ is a resonance precisely when the incoming coefficient at +∞ vanishes,
F(λ) = iλu(a) − u′(a) = 0. Roots are seeded on a grid and polished by the
secant method.
One honest limit. For a very long lived resonance the outgoing coefficient is
smaller than its companion by e−2∫κ, and recovering it from a
propagated solution is a cancellation of that size. Such resonances are tagged
long-lived: their real parts are accurate, their widths
are at the edge of what this formulation can resolve, and some may be missed entirely.
Recovering them accurately is precisely what the wave-trace method above does well,
since a narrow resonance is one that persists in time.
Accuracy. The potential is modelled as 400 constant slabs, for which the propagator
is exact; against a smooth reference the resonances agree to about 10−4.