The common-domain driven model: strong propagator with domain invariance, and the driven oscillator as its declared instance

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 common-domain class of clause (b)(iii): $\mathfrak{H}$ is the model's state space (Hilbert space), $Y\subseteq \mathfrak{H}$ is a dense linear subspace that is itself a Hilbert space continuously embedded in $\mathfrak{H}$, $D(H(t))=Y$ for every $t\in I=[a,b]$, every $H(t)$ is self-adjoint as an operator on $\mathfrak{H}$ (Symmetric, self-adjoint and essentially self-adjoint operators), and $t\mapsto H(t)$ is of class $C^1(I;\mathcal{B}(Y,\mathfrak{H}))$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum); put $A(t):=H(t)\mathrm{/}\mathrm{\hslash }$ and, when $a<b$, $L:={\sup }_{t\in I}\mathrm{\parallel }{H}^{\mathrm{\prime }}(t){\mathrm{\parallel }}_{\mathcal{B}(Y,\mathfrak{H})}<\mathrm{\infty }$; for $a=b$ set $L:=0$. 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). Then (all differential assertions below apply only when $a<b$, with one-sided derivatives at endpoints; for $a=b$ only $U(a,a)=I$ and the domain, norm and algebraic assertions are claimed):

(a) Well-posedness of the declared model. There is a family $U(t,s)$, $a\le s\le t\le b$ on $I$, of unitary operators on $\mathfrak{H}$ with $U(s,s)=I$, the cocycle $U(t,r)=U(t,s)U(s,r)$, joint strong continuity in $(t,s)$, and: $U(t,s)Y=Y,\mathrm{\parallel }U(t,s)y{\mathrm{\parallel }}_Y\le C_I\mathrm{\parallel }y{\mathrm{\parallel }}_Y(y\in Y),i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,s)y=H(t)U(t,s)y(y\in Y),$ where the derivative is in $\mathfrak{H}$ and $C_I<\mathrm{\infty }$ is independent of the endpoints and depends on the actual graph equivalence of $Y$ with $D(H(t)\mathrm{/}\mathrm{\hslash })$ as well as $I$ and $L\mathrm{/}\mathrm{\hslash }$; this family is unique among bounded families with the same strong equation on $Y$, $U(s,s)=I$ and locally bounded strong orbits. The model's clause (c)(iii) declaration therefore has exactly one solution.

(b) Reverse and adjoint equations. With $U(s,t):=U(t,s{)}^{-1}$ one has $U(t,s{)}^{\ast }=U(s,t)$ and $i\mathrm{\hslash }{\mathrm{\partial }}_{s}U(t,s)y=-U(t,s)H(s)y$ for $y\in Y$; equivalently $i\mathrm{\hslash }{\mathrm{\partial }}_{s}U(s,t)y=H(s)U(s,t)y$. The domain maps $(t,s)\mapsto U(t,s){\mathrm{\mid }}_Y$ are jointly strongly continuous in $\mathcal{B}(Y)$.

(c) Declared instance: the driven oscillator. Let $m,\omega ,\mathrm{\hslash }>0$ and let $(H_{\mathrm{o}\mathrm{s}\mathrm{c}},Q)$ be the energy and position observables of the one-dimensional oscillator model of The quadratic-potential oscillator model of the framework clauses (b), (c) on $L^2(\mathbb{R},dx)$, with $Y=D(H_{\mathrm{o}\mathrm{s}\mathrm{c}})$ carrying the graph norm, and let $f\in C^1(I;\mathbb{R})$. Declare, as a further model assumption of the pair, the driven oscillator family $H(t):=H_{\mathrm{o}\mathrm{s}\mathrm{c}}-f(t)Q(t\in I),$ read as the system's energy at time $t$, with $f(t)Q$ in joules for $f(t)$ in $\mathrm{J}{\mathrm{m}}^{-1}$. Then for every $t$ the operator $H(t)$ is self-adjoint on the single domain $Y$, the family belongs to $C^1(I;\mathcal{B}(Y,L^2))$ with derivative $-f^{\mathrm{\prime }}(t)Q$, and the hypotheses of clause (a) are satisfied with $\mathfrak{H}=L^2(\mathbb{R},dx)$; consequently the driven oscillator has a unique strong propagator preserving $D(H_{\mathrm{o}\mathrm{s}\mathrm{c}})$ and satisfying $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,s)y=H(t)U(t,s)y$ for $y\in D(H_{\mathrm{o}\mathrm{s}\mathrm{c}})$.

(d) Scope. The theorem is conditional on the common-domain regularity class and on the cited oscillator construction; it asserts no existence statement outside that class, no explicit solution, no long-time or adiabatic statement, no measurement and no empirical comparison.

• Why the common domain matters. The class-(iii) hypothesis gives what the bounded-control class of the mild theorem does not: invariance of the domain $Y$ and a genuine strong Schroedinger equation on $Y$. The price is the stronger regularity assumption, $C^1$ in the graph norm rather than mere strong continuity of a bounded control.
• The unbounded position coupling and constant drives. With the reference energy $H_0=H_{\mathrm{o}\mathrm{s}\mathrm{c}}$, the perturbation $-f(t)Q$ is unbounded whenever $f(t)\mathrm{\ne }0$, so general drives are treated here by the common-domain construction and relative bound of [F2]. A constant drive $f(t)\equiv f_0$ can instead be absorbed into the fixed self-adjoint energy $H_0=H_{\mathrm{o}\mathrm{s}\mathrm{c}}-f_{0}Q$ on $Y$ by [F2], with bounded control $V(t)=0$; in particular $f_0=0$ gives the free oscillator. Thus the regularity classes need not be disjoint. The explicit oscillator propagator is a separate result used by the page's example.