The bounded-control driven model: mild propagator, interaction-picture evolution and equivalence of the two readings

Adopt the framework data and the declared driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models in the bounded-control class of clause (b)(ii): $\mathfrak{H}$ is the model's state space, $H_0$ is self-adjoint on $D(H_0)$ (Symmetric, self-adjoint and essentially self-adjoint operators), $V:I\to \mathcal{B}(\mathfrak{H})$ is strongly continuous with self-adjoint values and $v:={\sup }_{t\in I}\mathrm{\parallel }V(t)\mathrm{\parallel }<\mathrm{\infty }$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum), $H(t)=H_0+V(t)$ on $D(H_0)$, and adopt the interaction-picture readings of The interaction picture of the declared driven model: readings, the interaction propagator and picture equivalence of outcome laws: $E(t)=e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}$, $V_I(t)=E(-t)V(t)E(t)$, $U_I(t,s)=E(-t)U(t,s)E(s)$. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), the latter for the outcome-law reading of P2. Let $U(t,s)$ be the mild propagator constructed by Strong bounded interaction picture. Then (all differential assertions apply only when $a<b$, with one-sided derivatives at endpoints; on $I=\{a\}$ the mild identity and all algebraic assertions reduce to $U(a,a)=U_I(a,a)=I$):

(a) Well-posedness of the declared model. $U$ is the unique family of unitary operators on $\mathfrak{H}$, jointly strongly continuous on $I\times I$, with $U(s,s)=I$, $U(t,s{)}^{\ast }=U(s,t)$ and the cocycle $U(t,r)=U(t,s)U(s,r)$, satisfying the vectorwise mild Schroedinger identity $U(t,s)\psi =E(t-s)\psi -\frac{i}{\mathrm{\hslash }}{\int }_s^{t}E(t-r)V(r)U(r,s)\psi dr(\psi \in \mathfrak{H},s,t\in I);$ so the model's clause (c)(ii) declaration has exactly one solution. For $\psi ,\phi \in D(H_0)$ the scalar curve $t\mapsto ⟨U(t,s)\psi ,\phi ⟩$ is continuously differentiable on $I$ with $\frac{d}{dt}⟨U(t,s)\psi ,\phi ⟩=-\frac{i}{\mathrm{\hslash }}[⟨U(t,s)\psi ,H_0\phi ⟩+⟨V(t)U(t,s)\psi ,\phi ⟩]\mathrm{.}$ No invariance of $D(H_0)$ under $U$, no strong full Schroedinger equation for its domain vectors, and no operator-norm differentiability of $U$ with respect to the unbounded generator are asserted.

(b) Interaction picture. For every $s,t\in I$ and every $\psi \in \mathfrak{H}$ the interaction state ${\psi }_I(t):=E(-t)U(t,s)\psi (s)$ satisfies ${\psi }_I(t)=U_I(t,s){\psi }_I(s)$, and its first-endpoint map is strongly differentiable with $i\mathrm{\hslash }{\mathrm{\partial }}_t{\psi }_I(t)=V_I(t){\psi }_I(t)$; the interaction propagator is unitary with the cocycle and $U_I(t,s{)}^{\ast }=U_I(s,t)$, and its strong truncation bound is $\mathrm{\parallel }{U}_I(t,s)-{\sum }_{n=0}^N{D}_n(t,s)\mathrm{\parallel }\le e^{K(t-s)}(K(t-s){)}^{N+1}\mathrm{/}(N+1)!$ for $s\le t$ with $K=v\mathrm{/}\mathrm{\hslash }$.

(c) Equivalence of the two readings. For every preparation $\rho$ specified at time $s$, every assigned sharp observable $(A,u_A)$ with $A$ bounded, every $t\in I$ and every Borel set $B\subseteq \mathbb{R}$, the Schroedinger reading and the interaction reading give the same outcome probability: ${\mu }_{\rho (t)}^A(B)={\mu }_{{\rho }_I(t)}^{A_I(t)}(B),A_I(t)=E(-t)AE(t),{\rho }_I(t)=E(-t)\rho (t)E(t),$ and for pure preparations $\mathrm{\parallel }{\psi }_I(t)\mathrm{\parallel }=\mathrm{\parallel }\psi (t)\mathrm{\parallel }=\mathrm{\parallel }\psi (s)\mathrm{\parallel }$. For eigenvectors ${\phi }_a,{\phi }_b$ of $H_0$ as in clause (e) of the model, the transition probabilities of the two pictures agree, $p_{ba}(t,s)=\mid ⟨U(t,s){\phi }_a,{\phi }_b⟩{\mid }^2=\mid ⟨U_I(t,s){\phi }_a,{\phi }_b⟩{\mid }^2\mathrm{.}$

(d) Scope. The theorem is conditional on the bounded-control class; the mild identity, not a strong equation on $D(H_0)$, is the evolution equation it establishes, and it asserts nothing about unbounded controls, invariant domains, measurement records, detectors, rates or empirical data.

• Why the mild equation and not more. The supplier deliberately does not claim that $U(t,s)$ maps $D(H_0)$ into itself, so the strong full Schroedinger equation for domain vectors is not available at this level of hypothesis; the weak identity of clause (a) is what remains, and it is enough for the amplitude theorem of this page because the eigenvector amplitudes are evaluated in the energy basis of $H_0$.
• The interaction picture does not change the model. Clause (c) records that $A$ and $\rho$ are transported together; a computation in the interaction picture is therefore a computation of the same outcome law, not a different physical situation.