Scattering, Resonance, and Linear Response

Guiding question: How are a system’s resonances encoded in its response to external forcing?

For a linear system, the retarded Green function is an inverse of the dynamical operator with causal boundary conditions. Isolated poles of a measured response identify freely evolving modes that couple to its chosen input and output: pole position gives oscillation frequency and decay or growth rate, while the residue measures that coupling. A mode with zero coupling can be absent from that response, and continuous spectra can contribute branch cuts instead of isolated poles. A real-frequency resonance peak may reveal a nearby pole, but its position also depends on the measured observable.

Required background. Ordinary Differential Equations provides initial-value problems, linearization, and asymptotic solution methods; Classical Field Theory provides field actions, Euler–Lagrange equations, and gauge redundancy. Helpful background. Calculus and Linear Algebra provides multivariable differentiation, linear maps, inner products, and basis changes; Partial Differential Equations and Green Functions provides characteristics, boundary data, distributions, and Green-function methods.

Scattering data and flux

Consider a one-dimensional lossless Sturm–Liouville problem

$$-\frac{d}{dx}\left(p(x)\frac{d\psi}{dx}\right) +V(x)\psi=\omega^2w(x)\psi, \qquad p,w\gt0,$$

with real coefficients that approach constants $p_a,V_a,w_a$ outside a finite region, where $a=L,R$ labels the left and right asymptotic regions. Restrict first to the two-open-channel regime, in which

$$p_ak_a^2+V_a=\omega^2w_a, \qquad k_a\gt0, \qquad a=L,R.$$

Thus both $k_L$ and $k_R$ are real propagating wavenumbers. For unit amplitude incident from the left, write

$$\psi_\omega(x)\sim \begin{cases} e^{ik_Lx}+R(\omega)e^{-ik_Lx},&x\to-\infty,\\ T(\omega)e^{ik_Rx},&x\to+\infty. \end{cases}$$

The weighted Wronskian flux

$$\boxed{\mathcal J=\frac{p(x)}{2i} \left(\psi_\omega^*\psi_\omega' -\psi_\omega\psi_\omega'^*\right)}$$

is constant because the coefficients are real. It equals $p_Lk_L(1-|R|^2)$ on the left and $p_Rk_R|T|^2$ on the right. Defining reflected and transmitted flux fractions by

$$\mathcal R=|R|^2, \qquad \mathcal T=\frac{p_Rk_R}{p_Lk_L}|T|^2,$$

gives

$$\boxed{\mathcal R+\mathcal T=1.}$$

Only when $p_Lk_L=p_Rk_R$ does this reduce to $|R|^2+|T|^2=1$. If either $k_a$ is imaginary, that asymptotic channel is evanescent and closed, so the displayed two-open-channel flux formula does not apply unchanged. In other wave normalizations the same factor is expressed through group velocity or impedance. Passive absorption instead gives $\mathcal R+\mathcal T\lt1$. Rosales derives the flux identity from a weighted Wronskian in Rosales 2023, §1.1, pp. 4–6 (official PDF).

An exactly solvable barrier checks the normalization. For

$$-\psi''+g\,\delta(x)\psi=k^2\psi, \qquad g\gt0,$$

continuity and the derivative jump give

$$1+R=T, \qquad \psi'(0^+)-\psi'(0^-)=g\psi(0).$$

Solving,

$$R=\frac{g}{2ik-g}, \qquad T=\frac{2ik}{2ik-g},$$

and hence

$$|R|^2=\frac{g^2}{4k^2+g^2}, \qquad |T|^2=\frac{4k^2}{4k^2+g^2}.$$

Their sum is one, low-frequency waves reflect, and high-frequency waves transmit. This concrete check also shows why amplitude conservation would be the wrong statement.

Reflection and transmission are real-frequency measurements. Analytically continuing the Green function or scattering amplitudes into complex frequency exposes bound-state or resonance poles, with the precise sheet and boundary condition part of the definition.

A response function with visible poles

For a driven damped oscillator,

$$\ddot q+2\gamma\dot q+\omega_0^2q=f(t), \qquad \gamma\gt0,\quad\omega_0\gt0,$$

use the convention $f,q\propto e^{-i\omega t}$. The steady response is

$$\widetilde q(\omega)=\chi(\omega)\widetilde f(\omega), \qquad \boxed{\chi(\omega) =\frac{1}{\omega_0^2-\omega^2-2i\gamma\omega}.}$$

Its poles are

$$\omega_\pm=-i\gamma \pm\sqrt{\omega_0^2-\gamma^2}.$$

They lie in the lower half-plane, so the retarded time response decays. For $0\lt\gamma\lt\omega_0$ they form an underdamped pair with common imaginary part $-\gamma$; at $\gamma=\omega_0$ they merge; for $\gamma\gt\omega_0$ they separate along the negative imaginary axis. In the weakly damped regime the pole real part is near $\omega_0$, the amplitude decay time is $\gamma^{-1}$, and the quality factor is $Q\simeq\omega_0/(2\gamma)$. MIT’s frequency-response notes derive gain and phase directly from the same characteristic polynomial in Orloff 2024, Topic 9, §§9.2–9.3, pp. 1–6 (official PDF).

Causality also constrains the entire real-frequency response, not only its poles. With

$$\chi(\omega)=\int_0^\infty dt\,e^{i\omega t}\chi(t),$$

$\chi(\omega)$ is analytic for $\operatorname{Im}\omega\gt0$. Because the oscillator response falls faster than $1/|\omega|$, the unsubtracted Kramers–Kronig relation is

$$\operatorname{Re}\chi(\omega) =\frac{1}{\pi}\,\mathop{\rm P}\!\int_{-\infty}^{\infty} \frac{\operatorname{Im}\chi(\omega')}{\omega'-\omega}\,d\omega'.$$

At zero frequency, reality makes $\operatorname{Im}\chi$ odd and gives the sign check

$$\chi(0)=\frac{2}{\pi}\int_0^\infty \frac{\operatorname{Im}\chi(\omega')}{\omega'}\,d\omega' =\frac{1}{\omega_0^2}\gt0.$$

The Fourier convention, analyticity domain, and dispersion relation are derived in Tong 2012, Kinetic Theory, §§4.1.2–4.1.3, pp. 81–86 (official PDF).

The maximum of $|\chi|$ occurs at

$$\omega_{\rm peak}=\sqrt{\omega_0^2-2\gamma^2}$$

when $\gamma\lt\omega_0/\sqrt2$. It is not exactly $\operatorname{Re}\omega_+$. Changing the measured observable—for example from displacement to velocity—can move the real-axis peak without moving the poles. Likewise, a pole can have a tiny residue and be nearly invisible to a poorly coupled probe.

For $\gamma\geq\omega_0/\sqrt2$, the displacement response has no nonzero-frequency maximum: $|\chi|$ is largest at $\omega=0$. Thus “resonance frequency” must name the observable and the regime.

Energy and interpretation

Multiply the oscillator equation by $\dot q$:

$$\frac{d}{dt} \left[\frac12\dot q^2+\frac12\omega_0^2q^2\right] =f\dot q-2\gamma\dot q^2.$$

Input power is divided between stored energy and nonnegative dissipation. In an open wave system, radiation to infinity supplies an analogous damping channel even when microscopic equations are conservative; an explicit string–oscillator realization is worked through in Rosales 2023, §2.2.3, pp. 13–14 (official PDF). A resonance width must therefore be interpreted together with all open channels and the normalization of flux.

Common failures are identifying every spectral bump with a pole, reading $|T|^2$ as transmitted power without flux weighting, putting a decaying mode in the upper half-plane under the $e^{-i\omega t}$ convention, or confusing a bound normal mode with an outgoing damped resonance. Upper-half-plane poles instead describe growing instabilities; a causal initial-value problem can be unstable, so retarded support alone is not a stability test.

Transfer task: drive at the natural frequency

For $f(t)=f_0\cos(\omega_0t)$, find the late-time amplitude and phase, and relate them to the nearest poles.

Solution

At $\omega=\omega_0$,

$$\chi(\omega_0)=\frac{i}{2\gamma\omega_0}.$$

Thus

$$q_{\rm steady}(t) =\frac{f_0}{2\gamma\omega_0}\sin(\omega_0t),$$

which lags the cosine drive by $\pi/2$. The amplitude grows as damping decreases but remains finite for $\gamma\gt0$. In the underdamped regime the poles share imaginary part $-\gamma$, so smaller $\gamma$ means slower ringdown and, when damping is weak, a narrower and taller response. At critical damping the two poles coincide at $-i\gamma$; in the overdamped regime they are purely imaginary at unequal depths. In both cases the language of a nearby oscillatory resonance is no longer appropriate. At $\gamma=0$, steady state fails at exact resonance and the amplitude grows linearly in time.

Continue to Thermodynamics, Statistical Mechanics, and Information, where susceptibilities, fluctuations, and dissipation are placed in a statistical framework.

References

• Jeremy Orloff, ES.1803 Topic 9: Applications—Frequency Response, MIT OpenCourseWare, 2024, §§9.2–9.4, pp. 1–8 (official PDF).
• R. R. Rosales, Lecture Topics for MIT 18.376, Massachusetts Institute of Technology, 2023, §1.1, pp. 4–7, and §§2.2.1–2.2.3, pp. 10–14 (official PDF).
• David Tong, Kinetic Theory, University of Cambridge, 2012, §§4.1.2–4.1.3, pp. 81–86 (official PDF).