Poles of the transmission amplitude

Solve the scattering problem the usual way and k appears only inside e^(±ikx) and inside linear matching conditions, all of which are entire functions of k. For a V of finite support every coefficient produced by the matching is therefore entire, and t(k) is a ratio of entire functions: one meromorphic function on the whole complex k plane, fixed by the same potential, with no extra assumption smuggled in. A potential that merely decays, like e^(−μ|x|), buys the continuation only above the line Im k = −μ/2; a Coulomb tail breaks it altogether and needs a different frame. The real positive axis, where k = √(2mE)/ħ, is one line drawn across that object. Every bound level, every resonance and every virtual state the potential supports is encoded in where t stops being finite away from that line. The move of this topic is to stop treating negative and positive energies as two calculations and treat them as two regions of one function.

Write ψ = e^(ikx) + r e^(−ikx) on the left and t e^(ikx) on the right. Every amplitude has already been divided by the incoming one, so t = ∞ says a finite outgoing wave survives a vanishing input. Formalise it with the Jost solutions f₊(k, x) → e^(ikx) as x → +∞ and f₋(k, x) → e^(−ikx) as x → −∞, whose Wronskian gives t(k) = 2ik/W(k). A pole is a zero of W, and a vanishing Wronskian means f₊ and f₋ are the same solution up to a constant: one function, purely outgoing at both ends. That is the Siegert boundary condition, and it is the only place in this unit where a boundary condition is imposed at complex k. Read it as a resonator condition — the potential ringing on its own — and bound states and resonances stop being two problems. They are the same question asked in two regions of one plane.

Put k = iκ. Then e^(ikx) = e^(−κx) dies as x → +∞ and e^(−ikx) = e^(+κx) dies as x → −∞, so for κ > 0 the outgoing-only solution is square-integrable: a bound state at E = ħ²k²/2m = −ħ²κ²/2m, below the asymptotic floor. A short-range V supports finitely many, and Levinson's theorem counts them from the phase of t: in one dimension, for a generic V with no zero-energy resonance, arg t(0⁺) − arg t(∞) = (n_b − ½)π, half a unit short of the three-dimensional s-wave count n_b π. Everywhere else the same solution leaves L². On the negative imaginary axis, k = −iκ, it grows at both ends: a virtual or antibound state, no eigenvector at all, yet a strong influence on low-energy scattering — the neutron-proton singlet pole sits there and produces the anomalous scattering length aₛ = −23.7 fm. Reality of V forces t(−k*) = t(k)*, so a pole either sits on the imaginary axis or comes with its mirror at −k*; and for a fast-decaying V no pole is permitted in the open upper half plane off that axis, since it would be a state growing in time.

A pole at k = kᵣ − iγ with kᵣ, γ > 0 squares to E = (ħ²/2m)(kᵣ² − γ² − 2i kᵣγ). Call that E_R − iΓ/2 and you have E_R = ħ²(kᵣ² − γ²)/2m and Γ = 2ħ²kᵣγ/m: peak position and full width, from two coordinates. Because E = ħ²k²/2m is two-to-one, the k plane double-covers the E plane — the upper half is the physical sheet, and the lower half, where every resonance lives, is the second sheet, reached by circling the branch point at E = 0. That is the whole reason a Hermitian H can carry a complex number here without contradiction. Take an electron pole at k = (0.80 − 0.02i) Å⁻¹ with ħ²/2m = 3.810 eV⋅Ų: E_R = 3.810 × 0.6396 = 2.44 eV, Γ = 3.810 × 0.0640 = 0.244 eV, and the γ² term moves E_R by only 1.5 meV. Near an isolated pole T(E) collapses onto the Breit–Wigner form, which is why a fitted peak is a pole measurement.

Attach the time factor. e^(−iEt/ħ) with E = E_R − iΓ/2 gives |ψ(t)|² ∝ e^(−Γt/ħ), so τ = ħ/Γ: Γ = 0.244 eV is 2.70 fs, Γ = 0.400 meV is 1.65 ps, and the 117 MeV width of the Δ(1232) is 5.6×10⁻²⁴ s. The same complex k makes the spatial part diverge, since |e^{(ikᵣx + γx)}| = e^(γx) blows up as x → +∞. So the Siegert or Gamow state is not a vector in L², and E_R − iΓ/2 is not an eigenvalue of the self-adjoint H. The growth is bookkeeping, not paradox: the amplitude now sitting at distance x left the well a time x/v ago, when the state was more populated by exactly that factor. A physical packet truncates the tail at the causal radius x ≈ vt. Exponential decay is itself only asymptotic — quadratic at very short times, and power-law at very long ones, where the branch cut at E = 0 outlives the pole.

Route one: build t(k) from a transfer or scattering matrix and Newton-iterate on 1/t(k) = 0 in complex k, seeded from a Breit–Wigner fit of the real-axis peak. Cheap, but the transfer-matrix product conditions like e^(2κL), so a thick barrier destroys the root before the iteration converges; use the stable scattering recursion instead. Route two: complex scaling. Rotate x → x e^(iθ), and the continuum swings down by 2θ about E = 0 while any resonance inside that wedge becomes a genuine square-integrable eigenvector of the rotated non-Hermitian Hamiltonian, so scipy.linalg.eig on a rotated finite-difference matrix returns E_R − iΓ/2 directly. Route three: fit T(E) on a real grid. Fast and honest for one isolated narrow pole, and wrong the moment two resonances overlap or a background phase tilts the line into a Fano profile — then only the continuation gives an unambiguous E_R and Γ.