Spectral survival recurrence

A specified evolution and preparation

Let $H$ be a self-adjoint frequency operator and $u$ a normalized field profile. Time evolution is $u(t)=e^{-itH}u$; for an energy operator replace $H$ by $H/\hbar$. The return amplitude and weight are

$$a(t)=\langle u,e^{-itH}u\rangle, \qquad P(t)=|a(t)|^2.$$

These are Hilbert-space diagnostics of the selected linear evolution. They apply to classical complex mode amplitudes as well as quantum states. Their interpretation as an observed survival probability requires a measurement prescription. Mode amplitudes describe fields on the one physical manifold and introduce no further spatial dimensions.

For a preparation supported on the pure-point spectrum, combine each degenerate eigenspace into one orthogonal projector $P_j$, with distinct frequencies $\omega_j$. Put $p_j=\|P_ju\|^2$. Then

$$p_j\geq0,\quad \sum_jp_j=1, \qquad a(t)=\sum_j p_j e^{-i\omega_jt}.$$

The series converges absolutely and uniformly in time, because its omitted tail is bounded by the omitted sum of $p_j$.

Time average

For a finite spectral sum, integrate each pair of phases:

$$\frac1T\int_0^T e^{-i(\omega_j-\omega_k)t}\,dt \longrightarrow \begin{cases}1&j=k,\\0&j\ne k.\end{cases}$$

Uniform tail control extends this calculation to a countable sum. Hence

$$\boxed{\lim_{T\to\infty}\frac1T\int_0^T P(t)\,dt =\sum_jp_j^2>0.}$$

Degeneracies must be grouped before squaring their weights. The average of $a(t)$ itself retains only a zero-frequency component, so it is a different quantity. A small mean return weight can coexist with eventual recurrence.

For two equally weighted frequencies,

$$a(t)=e^{-i(\omega_1+\omega_2)t/2} \cos\!\left(\frac{\omega_2-\omega_1}{2}t\right), \qquad P(t)=\cos^2\!\left(\frac{\omega_2-\omega_1}{2}t\right).$$

The recurrence of the ray occurs every $2\pi/|\omega_2-\omega_1|$. The difference of frequencies controls the beating.

Approximate recurrence without commensurability

Choose a finite set of occupied frequencies and a reference $\omega_1$. For any time unit $\tau>0$, consider the phase vectors $k\tau(\omega_j-\omega_1)/(2\pi)$ modulo one, for integers $k$. Partition this finite-dimensional unit cube into boxes of side $1/Q$. Among one more points than boxes, two share a box. Subtracting their integer times gives a positive $t$ at which every selected relative phase is within $2\pi/Q$ of an integer multiple of $2\pi$.

For a countable spectrum choose the finite set with omitted weight $\eta$. If the selected phase errors have absolute value at most $\epsilon$,

$$\|e^{-itH}u-e^{-i\omega_1t}u\|^2 =\sum_jp_j|e^{-i(\omega_j-\omega_1)t}-1|^2 \leq\epsilon^2+4\eta.$$

Both terms can be made arbitrarily small. Such times are unbounded: if the approximating integer times remained bounded as the tolerance shrank, one of them would be an exact common period, whose multiples are unbounded. For any fixed tail cutoff the same argument supplies arbitrarily late returns. Exact ray recurrence requires all occupied frequency differences times $t$ to lie in $2\pi\mathbb Z$; approximate recurrence needs no such commensurability. This is the spectral recurrence setting discussed by Campos Venuti, §I.

An absolutely continuous preparation

The spectral theorem more generally gives a probability measure $\mu_u$:

$$a(t)=\int e^{-i\omega t}\,d\mu_u(\omega).$$

If $d\mu_u=f(\omega)d\omega$ with $f\in L^1$, the Riemann–Lebesgue lemma gives $a(t)\to0$. For a uniform density on $[\omega_0-B,\omega_0+B]$, $B>0$,

$$a(t)=e^{-i\omega_0t}\frac{\sin Bt}{Bt}.$$

This decay describes dephasing of the return amplitude while the full evolution remains unitary. Singular continuous spectral measures require their own analysis; absolute continuity is the sufficient hypothesis used here. Selecting a physical persistent mode additionally requires its coupling and dynamics, beyond a recurrence identity.

Verification and related

spectral_analysis_checks.py checks exact finite time averages, commensurate and incommensurate examples, and continuous-density Fourier integrals.

• Continuum mode memory — derives how coupling to a continuum changes the reduced amplitude equation.
• Confining oscillator domain — supplies one spatial operator with a pure-point spectrum.
• Spectral analysis tools — navigation and scope.