Explicit unbounded oscillator propagator

Schwartz functions belong to $Y$ and form a graph core. Indeed, for $\phi \in \mathcal{S}(\mathbb{R})$ the differential expression $L\phi$ is in $L^2$; integration by parts with the Schwartz eigenfunction ${\psi }_n$ gives $⟨L\phi ,{\psi }_n⟩=E_n⟨\phi ,{\psi }_n⟩$. Parseval therefore shows ${\sum }_n{E}_n^2\mathrm{\mid }⟨\phi ,{\psi }_n⟩{\mathrm{\mid }}^2=\mathrm{\parallel }L\phi {\mathrm{\parallel }}^2<\mathrm{\infty }$, proving $\phi \in Y$ and $H_{\mathrm{o}\mathrm{s}\mathrm{c}}\phi =L\phi$ by the exact diagonal domain. Finite Hermite sums are Schwartz and approximate every $Y$ vector in graph norm by [F1], so Schwartz is a graph core. On finite sums the ladder norm estimate in [F2] bounds $Q,P$ by a constant times the $Y$ norm, since $\mathrm{\parallel }{N}^{1\mathrm{/}2}\psi {\mathrm{\parallel }}^2\le \mathrm{\parallel }N\psi \mathrm{\parallel }\mathrm{\parallel }\psi \mathrm{\parallel }$ and $N=(\mathrm{\hslash }\omega {)}^{-1}{H}_{\mathrm{o}\mathrm{s}\mathrm{c}}-I\mathrm{/}2$. Closedness of $Q,P$ extends these graph bounds and their actions to all of $Y$.

Define $F(\tau )u={\sum }_n{e}^{-i\tau E_n\mathrm{/}\mathrm{\hslash }}{c}_n{\psi }_n$ for $c_n=⟨u,{\psi }_n⟩$. Parseval proves convergence, unit norm, inverse $F(-\tau )$, and the group law by coefficient multiplication. Strong continuity follows by splitting ${\sum }_n\mathrm{\mid }{e}^{-i\tau E_n\mathrm{/}\mathrm{\hslash }}-e^{-i\sigma E_n\mathrm{/}\mathrm{\hslash }}{\mathrm{\mid }}^2\mathrm{\mid }{c}_n{\mathrm{\mid }}^2$ into a finite sum tending to zero and a tail bounded by $4{\sum }_{n>M}\mathrm{\mid }{c}_n{\mathrm{\mid }}^2$. For $u\in Y$, the identical argument with weights $1+E_n^2$ gives graph continuity and $F(\tau )Y=Y$. Integrating the derivative of the scalar exponential gives $\mathrm{\mid }{e}^{-ix}-1\mathrm{\mid }\le \mathrm{\mid }x\mathrm{\mid }$, so the difference-quotient error coefficients are bounded by $2E_n\mathrm{\mid }{c}_n\mathrm{\mid }\mathrm{/}\mathrm{\hslash }$ and tend individually to zero. Their square-summable majorant gives the strong derivative ${\mathrm{\partial }}_{\tau }F(\tau )u=-iAF(\tau )u\mathrm{/}\mathrm{\hslash }$ by the same finite-sum/tail argument. Conversely a norm derivative at zero must have coefficients $-i{E}_n{c}_n\mathrm{/}\mathrm{\hslash }$, so Parseval forces $u\in Y$. We write this explicitly defined diagonal group as $e^{-i\tau A\mathrm{/}\mathrm{\hslash }}$. No general spectral calculus or AC-bearing unitary-group theorem is used.

The classical path can be defined without an unproved generic ODE existence invocation: $q(t;s)=(m\omega {)}^{-1}{\int }_s^t\sin (\omega (t-r))f(r)dr$ and $p(t;s)={\int }_s^t\cos (\omega (t-r))f(r)dr$. Differentiating these continuous integrals gives $q^{\mathrm{\prime }}=p\mathrm{/}m$, $p^{\mathrm{\prime }}=-m{\omega }^{2}q+f$ and the zero initial data. If two real paths solve the system, the difference satisfies the homogeneous equations and its nonnegative energy $p^2\mathrm{/}(2m)+m{\omega }^2{q}^2\mathrm{/}2$ has derivative zero; its zero initial value forces both differences to vanish. The formulas give joint continuity in $(t,s)$ on the compact triangle, and $S^{\mathrm{\prime }}=f(t)q(t;s)\mathrm{/}2$ follows directly from its defining integral.

The operator squares $Q^2=M_{x^2}$ and $P^2={\mathcal{F}}_2^{\ast }{M}_{(2\pi \mathrm{\hslash }\xi {)}^2}{\mathcal{F}}_2$ are closed on their maximal multiplier domains: apply the concrete real-multiplier adjoint argument from the common-domain supplier to these symbols and transport the second by Plancherel. The square domains agree with the operator-square domains, since $\mathrm{\mid }r\mathrm{\mid }\le 1+r^2$ places a vector in the first multiplier domain whenever it is in the squared multiplier domain. On finite Hermite sums, the ladder actions give $\mathrm{\parallel }{a}_{\psi }^{2}u{\mathrm{\parallel }}^2={\sum }_{n}n(n-1)\mathrm{\mid }{c}_n{\mathrm{\mid }}^2\le \mathrm{\parallel }{N}_{\psi }u{\mathrm{\parallel }}^2$, $\mathrm{\parallel }(a_{\psi }^{†}{)}^{2}u{\mathrm{\parallel }}^2={\sum }_n(n+1)(n+2)\mathrm{\mid }{c}_n{\mathrm{\mid }}^2\le 2\mathrm{\parallel }(N_{\psi }+I)u{\mathrm{\parallel }}^2$, and the mixed products are $N_{\psi }$ and $N_{\psi }+I$. Expanding $(a_{\psi }\pm a_{\psi }^{†}{)}^2$ and using the triangle inequality gives its norm at most $(3+\sqrt{2})\mathrm{\parallel }(N_{\psi }+I)u\mathrm{\parallel }\le 6\mathrm{\parallel }(N_{\psi }+I)u\mathrm{\parallel }$. Thus $\mathrm{\parallel }{Q}^{2}u\mathrm{\parallel }\le 3{\mathrm{\ell }}^2\mathrm{\parallel }(N_{\psi }+I)u\mathrm{\parallel }$ and $\mathrm{\parallel }{P}^{2}u\mathrm{\parallel }\le 3{\mathrm{\hslash }}^2{\mathrm{\ell }}^{-2}\mathrm{\parallel }(N_{\psi }+I)u\mathrm{\parallel }$. Since $n+1\le 2(n+1\mathrm{/}2)$, these are $Y$ graph bounds. Finite coefficient truncations of any $u\in Y$ therefore have convergent square-operator images; closedness yields $Y\subseteq D(Q^2)\cap D(P^2)$. On the finite sums the differential identity in [F1] and the Fourier derivative identity in [F3] give $Au=P^{2}u\mathrm{/}(2m)+m{\omega }^2{Q}^{2}u\mathrm{/}2$; the graph limits extend this identity to all $Y$. This justifies the exact quadratic sum in the Statement.

The Weyl formula is translation followed by multiplication by a unimodular phase, so it is unitary with inverse $D(-q,-p)=D(q,p{)}^{\ast }$. It maps Schwartz onto Schwartz. Differentiation and multiplication there give $QD=D(Q+qI)$ and $PD=D(P+pI)$, hence $H_{\mathrm{o}\mathrm{s}\mathrm{c}}D\phi =D(H_{\mathrm{o}\mathrm{s}\mathrm{c}}+(p\mathrm{/}m)P+m{\omega }^{2}qQ+E(q,p)I)\phi$, where $E(q,p)=p^2\mathrm{/}(2m)+m{\omega }^2{q}^2\mathrm{/}2$. Given $\psi \in Y$, take its finite Hermite partial sums ${\phi }_n\to \psi$ in graph norm. By step 1.1 the right-hand side of this identity converges in $L^2$ and $D{\phi }_n\to D\psi$; closedness of $H_{\mathrm{o}\mathrm{s}\mathrm{c}}$ gives $D\psi \in Y$ and the same identity there. Applying this inclusion to the inverse gives $D(Y)=Y$. The corresponding $Q,P$ identities extend by their closedness using the same approximants, proving the two asserted conjugations. In particular $D$ is bounded $Y\to Y$, locally uniformly in $(q,p)$. Moving $D{H}_{\mathrm{o}\mathrm{s}\mathrm{c}}$ to the left in the last identity and replacing $DP$ by $(P-pI)D$, $DQ$ by $(Q-qI)D$, gives the output-sided identity $H_{\mathrm{o}\mathrm{s}\mathrm{c}}D=D{H}_{\mathrm{o}\mathrm{s}\mathrm{c}}+[(p\mathrm{/}m)P+m{\omega }^{2}qQ-E(q,p)I]D$ on $Y$. Both scalar energy terms have minus signs in this form.

For a fixed $\psi \in Y$, the map $(q,p)\mapsto D(q,p)\psi$ is continuous in $Y$: its $L^2$ continuity is [F3], and its $H_{\mathrm{o}\mathrm{s}\mathrm{c}}$ image is the continuous expression $D(q,p)[H_{\mathrm{o}\mathrm{s}\mathrm{c}}+(p\mathrm{/}m)P+m{\omega }^{2}qQ+E(q,p)I]\psi$ from step 2.2. On a $C^1$ path differentiate the Weyl formula on Schwartz, where ordinary dominated differentiation is valid. With $PD=D(P+pI)$ this gives $i\mathrm{\hslash }{D}^{\mathrm{\prime }}\phi =[-p^{\mathrm{\prime }}Q+q^{\mathrm{\prime }}P+(q{p}^{\mathrm{\prime }}-p{q}^{\mathrm{\prime }})I\mathrm{/}2]D\phi$. To extend rigorously, approximate $\psi$ by Schwartz graph-core vectors ${\phi }_n$. On a compact time interval, the local uniform graph bound in step 2.2 and bounded $Q,P:Y\to L^2$ show that both $D(t){\phi }_n\to D(t)\psi$ and the proposed derivatives converge uniformly in $L^2$. For each $n$ integrate the derivative identity between a fixed base time and $t$ by [F3]; uniform convergence permits taking the integral limit. The limiting right-hand side is continuous, by the $Y$ continuity just proved and continuity of $q^{\mathrm{\prime }},p^{\mathrm{\prime }}$. Applying [F3] again proves the strong derivative for $\psi$ and the scalar $(q{p}^{\mathrm{\prime }}-p{q}^{\mathrm{\prime }})\mathrm{/}2$. This proves all of (a), including its domain and differentiability assertions.

Put $A=H_{\mathrm{o}\mathrm{s}\mathrm{c}}$, $F(t)=e^{-i(t-s)A\mathrm{/}\mathrm{\hslash }}$, $c_D=(q{p}^{\mathrm{\prime }}-p{q}^{\mathrm{\prime }})\mathrm{/}2$ and $X=(p\mathrm{/}m)P+m{\omega }^{2}qQ-E(q,p)I$. The unitary family $U=e^{iS\mathrm{/}\mathrm{\hslash }}DF$ preserves $Y$ by steps 2.2 and 1.2. Its product derivative on $Y$ is legitimate: for fixed $\psi \in Y$ expand $D(t+h)F(t+h)\psi -D(t)F(t)\psi$ as $D(t+h)[F(t+h)-F(t)]\psi +[D(t+h)-D(t)]F(t)\psi$. The first quotient tends to $D(t)(-iA\mathrm{/}\mathrm{\hslash })F(t)\psi$ by free strong differentiation from Step 1.2, strong continuity and unit norm of $D$; the second tends to $D^{\mathrm{\prime }}(t)F(t)\psi$ by step 3.1 for the fixed vector $F(t)\psi \in Y$. Thus $i\mathrm{\hslash }{U}^{\mathrm{\prime }}=e^{iS\mathrm{/}\mathrm{\hslash }}[A-X-S^{\mathrm{\prime }}-p^{\mathrm{\prime }}Q+q^{\mathrm{\prime }}P+c_{D}I]DF$ on $Y$. The $P$ coefficient is $q^{\mathrm{\prime }}-p\mathrm{/}m=0$, the $Q$ coefficient is $-p^{\mathrm{\prime }}-m{\omega }^{2}q=-f$, and the scalar is $-S^{\mathrm{\prime }}+c_D+E(q,p)=-fq\mathrm{/}2+[q(f-m{\omega }^{2}q)-p^2\mathrm{/}m]\mathrm{/}2+p^2\mathrm{/}(2m)+m{\omega }^2{q}^2\mathrm{/}2=0$. Hence the unchanged final phase $S^{\mathrm{\prime }}=fq\mathrm{/}2$ gives $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U=H(t)U$. Step 3.1, the graph-continuous free factor of Step 1.2 and the locally uniform graph bounds also imply that $U(t,s)\psi$ is continuous in $Y$, so the displayed derivative is continuous in $L^2$ because $H(t):Y\to L^2$ is continuous. Finally zero initial data give $U(s,s)=I$.

Let $v(t)$ and $w(t)$ be two $Y$-valued strong solutions with the same initial vector. Their difference $z$ is $L^2$-norm differentiable and solves $z^{\mathrm{\prime }}=-iH(t)z\mathrm{/}\mathrm{\hslash }$. The norm derivative is ${\mathrm{\partial }}_t\mathrm{\parallel }z(t){\mathrm{\parallel }}^2=2\mathrm{Re}⟨z^{\mathrm{\prime }},z⟩=2\mathrm{Re}[-i⟨H(t)z,z⟩\mathrm{/}\mathrm{\hslash }]=0$, because $H(t)$ is symmetric and the expectation is real. Since $z(s)=0$, $z=0$. This proves uniqueness without a bound on an unbounded $L^2$ Hamiltonian or an invalid norm-factorial iteration. For $r\le s\le t$, both $U(t,r)\psi$ and $U(t,s)U(s,r)\psi$ solve the same strong equation with initial vector $U(s,r)\psi \in Y$ at $s$, so uniqueness gives the cocycle on $Y$, then on $L^2$ by density. Joint strong continuity on $L^2$ follows from the explicit continuous classical paths and phase, free strong continuity and unitarity. Since $Y$ is dense, graph-domain solutions approximating an arbitrary initial $L^2$ vector converge uniformly in $t$ in $L^2$, with error equal to the initial norm error; this is the asserted continuous mild evolution. The inverses are unitary and preserve $Y$ as well; for all time orders use the same explicit classical formulas and phase with oriented integrals. Their differentiations hold on either side of the initial time, so strong uniqueness gives the cocycle for any intermediate time. In particular $U(t,s)U(s,t)=I$, giving $U(s,t)=U(t,s{)}^{\ast }$.

If $f=0$, step 1.3 gives $q=p=S=0$, so $U(t,s)=e^{-i(t-s)H_{\mathrm{o}\mathrm{s}\mathrm{c}}\mathrm{/}\mathrm{\hslash }}$. The coherent series converges and has unit norm because its coefficient norm-square is $e^{-\mathrm{\mid }z{\mathrm{\mid }}^2}{\sum }_n\mathrm{\mid }z{\mathrm{\mid }}^{2n}\mathrm{/}n!=1$. For every fixed integer $k$, the ratio of successive positive terms $n^{2k}\mathrm{\mid }z{\mathrm{\mid }}^{2n}\mathrm{/}n!$ tends to zero for $z\mathrm{\ne }0$, so its tail is bounded by a geometric series; $z=0$ is immediate. Thus the vector belongs to $D(N_{\psi }^k)$, in particular $Y$, and is precisely the scaled coherent vector in the Statement. The diagonal coefficient formula from Step 1.2 gives $U(t,s){\psi }_z=e^{-i\omega (t-s)\mathrm{/}2}{\psi }_{z{e}^{-i\omega (t-s)}}$. Thus (b), (c), the exact propagator and all original domain/range assertions hold with the corrected Weyl identities.