Theorem: Prepared spectral measure and finite-time decay

For (eq:semig), the prepared source spectral measure is exactly

It has no pure-point or singular-continuous component. In particular, the prepared code has no bound-state component.

The survival amplitude is

where

The value at $T=0$ is one. For every $T>0$,

The emitted wavefunction is (eq:fieldsolution).

Proof. For $z\in \mathbb{C}\setminus \mathbb{R}$, set

Substitution $x=2c\cos \theta$ and the elementary Cauchy integral evaluation give

with branch $s(z)\sim z-{\omega }_0$ at infinity. Equivalently,

The Schur complement is valid for nonreal $z$ and gives

Uniqueness of the measure from its Cauchy transform identifies the entire prepared spectral measure. This argument does not declare real frequencies in the uncoupled continuum to be resolvent points. It proves the absence of hidden atoms and singular components in the prepared measure.

The spectral theorem gives (eq:exactA). The even moments are

and the odd moments vanish. Taylor expansion on the compact interval proves the series in (eq:bessel), which is the standard $J_1$ series DLMF (Section 10.2).

The zero extension of $w_c$ is absolutely continuous, vanishes at the endpoints, and satisfies

Integration by parts gives $|a_c(T)|\le 2/(\pi cT)$. Combine this with unitarity.