Continuum mode memory

Coupled amplitudes

Consider a discrete amplitude $a$ coupled to amplitudes $b(\omega)$ with measure $d\nu(\omega)$. In frequency units let

$$\begin{aligned} i\dot a&=\omega_0a+\int g(\omega)b(\omega)\,d\nu(\omega),\\ i\dot b(\omega)&=\omega b(\omega)+g(\omega)^*a. \end{aligned}$$

Assume $\omega_0$ is real, multiplication by $\omega$ is self-adjoint on its natural $L^2(d\nu)$ domain, and $g\in L^2(d\nu)$. The off-diagonal coupling is then bounded and the full operator on $\mathbb C\oplus L^2(d\nu)$ is self-adjoint. It conserves

$$|a|^2+\int|b(\omega)|^2\,d\nu(\omega).$$

The two coupling contributions cancel upon differentiation. These are coefficients of field modes in the nested physical manifold; the spectral labels are not extra physical spatial directions.

Exact elimination

Solving the second equation by an integrating factor gives

$$b(\omega,t)=e^{-i\omega t}b(\omega,0) -ig(\omega)^*\int_0^t e^{-i\omega(t-s)}a(s)\,ds.$$

Substitution into the first gives the exact causal equation

$$\boxed{\dot a(t)=-i\omega_0a(t)-iF(t) -\int_0^t K(t-s)a(s)\,ds,}$$

where

$$F(t)=\int g(\omega)e^{-i\omega t}b(\omega,0)\,d\nu(\omega), \qquad K(t)=\int|g(\omega)|^2e^{-i\omega t}\,d\nu(\omega).$$

An initially unexcited continuum has $F=0$. An occupied continuum supplies a forcing term and cannot be represented by the same homogeneous equation.

For $\operatorname{Re}z>0$, the zero-forcing Laplace transform satisfies

$$\widetilde a(z)= \frac{a(0)}{z+i\omega_0+\displaystyle\int \frac{|g(\omega)|^2}{z+i\omega}\,d\nu(\omega)}.$$

The frequency dependence in this denominator retains the response of the eliminated modes. Poles and continuum contributions both affect evolution.

Weak-coupling decay approximation

Suppose $d\nu=\rho(\omega)d\omega$ and write $J(\omega)=\rho(\omega)|g(\omega)|^2$. With $A(t)=e^{i\omega_0t}a(t)$,

$$\dot A(t)=-\int_0^t k(\tau)A(t-\tau)\,d\tau, \qquad k(\tau)=\int J(\omega)e^{-i(\omega-\omega_0)\tau}\,d\omega.$$

If the kernel loses memory much faster than $A$ changes, replace $A(t-\tau)$ by $A(t)$ and the upper limit by infinity. This approximation requires weak coupling, adequate decay of the kernel, a smooth density near an interior resonance, and separation from relevant spectral edges or bound-state poles. Insert $e^{-\epsilon\tau}$ before taking $\epsilon\downarrow0$:

$$\int_0^\infty k(\tau)\,d\tau =\pi J(\omega_0)+i\,\operatorname{PV}\!\int \frac{J(\omega)}{\omega_0-\omega}\,d\omega.$$

This follows from the distributional limit of $[\epsilon+i(\omega-\omega_0)]^{-1}$. If the principal value exists,

$$a(t)\approx a(0)e^{-i(\omega_0+\Delta)t-\Gamma t/2}, \quad \Delta=\operatorname{PV}\!\int\frac{J(\omega)}{\omega_0-\omega}\,d\omega, \quad \Gamma=2\pi J(\omega_0).$$

$\Gamma$ is the decay rate of $|a|^2$; the amplitude rate is half as large. The approximation need not describe very short times, late tails, or a surviving discrete component of the full spectrum.

Finite and continuous benchmarks

For a single resonant companion with real coupling $g$, initial data $(a,b)=(1,0)$ give

$$a(t)=e^{-i\omega_0t}\cos(gt), \qquad b(t)=-ie^{-i\omega_0t}\sin(gt).$$

The transfer reverses periodically. A finite set of companions has the spectral recurrences of a finite self-adjoint matrix. More companions can approximate a continuum on a finite time interval, with a recurrence scale that must be checked.

For constant $J=J_0$ on $[\omega_0-B,\omega_0+B]$ and zero elsewhere, the exact rotating kernel is $k(t)=2J_0\sin(Bt)/t$, with $k(0)=2BJ_0$. The memory equation retains its finite bandwidth. This supplies a controlled comparison for an exponential approximation without postulating a frequency filter on $\Psi_\infty$.

Verification and related

spectral_analysis_checks.py compares finite matrix evolution with the eliminated convolution equation, checks conserved norm and Laplace response, and integrates the finite-band kernel independently.

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