Section 4 4 October 2026

Exact Gaussian returns with fixed interactions

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131064

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4 Exact Gaussian returns with fixed interactions

This section separates two endpoint problems. The classical symplectic propagator determines exact quantum return for a quadratic Hamiltonian. The configuration propagator is computed from its evolving Gaussian phase. They are different matrices.

4.1 Symplectic endpoint correction

Let $K_0=O\Omega^2O^T>0$ , with $\Omega=\diag(\omega_1,\ldots,\omega_d)$ . First suppose the frequencies are positive integers. At $T=2\pi$ the uncontrolled symplectic propagator is the identity. Define

\dot{\mathsf S}=\mathsf A(K_0+D(t))\mathsf S,\qquad \mathsf A(K)=\begin{pmatrix}0&I\\-K&0\end{pmatrix},\qquad \mathsf S(0)=I .

(4.1)

Here $\mathsf S$ acts on classical position and momentum, used to represent the quadratic quantum evolution. It is not a guidance flow. The metaplectic representation, or equivalently the exact Heisenberg equations for Weyl operators, shows that $\mathsf S(T)=I$ implies $U(T)=e^{i\theta}I$ on the full quantum Hilbert space [5, Chapter 4]. Irreducibility of the Weyl representation gives the same conclusion directly. There is no spectral truncation in this implication.

For a physical matrix $M\in\calD$ write $\widetilde M=O^TMO$ . The following finite-dimensional criterion will be arranged by engineering the holding traps.

Lemma 4.1 (Endpoint submersion)

Assume that

(i)
the positive numbers $2\omega_i$ , $\omega_i+\omega_j$ and $|\omega_i-\omega_j|$ ( $i<j$ ) are pairwise distinct;
(ii)
for each $i<j$ some $M\in\calD$ has $\widetilde M_{ij}\ne0$ ;
(iii)
the vectors $(\widetilde M_{11},\ldots,\widetilde M_{dd})$ , $M\in\calD$ , span $\R^d$ .

Then the endpoint map from smooth, interior-supported physical quadratic controls to $\Sp(2d,\R)$ is a submersion at zero control and time $2\pi$ . A finite-dimensional family of such controls already has this property.

Proof

At the return time the differential is

D\mathcal E(0)[D]=\int_0^{2\pi} \mathsf S_0(t)^{-1} \begin{pmatrix}0&0\\-D(t)&0\end{pmatrix} \mathsf S_0(t)\,dt .

(4.2)

In oscillator coordinates put $q_i(t)=(a_i e^{-i\omega_i t}+\bar a_i e^{i\omega_i t})/ \sqrt{2\omega_i}$ . A real quadratic $q^TMq/2$ then has the two real quadratures of $a_ia_j$ at $\omega_i+\omega_j$ , those of $a_i\bar a_j$ at $\omega_i-\omega_j$ , and the diagonal number terms at frequency zero. For $i=j$ the nonzero frequency is $2\omega_i$ .

If a real covector on $\mathfrak{sp}(2d,\R)$ annihilates (4.2) for every interior-supported time waveform and every $M$ , its pairing with this quadratic expression vanishes identically in $t$ . Independence of the distinct trigonometric frequencies, (ii), and (iii) force its coefficients on every root quadrature and on every diagonal number term to vanish. These form a real basis of $\mathfrak{sp}(2d,\R)$ , so the differential is onto. Select $d(2d+1)$ smooth controls whose images are a basis. Their coefficients give the asserted finite-dimensional submersion.

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Choose two distinct positive integers $\nu,\zeta$ larger than every frequency occurring in (i). For $M,N\in\calD$ the trial control

D_{\mathrm{tr}}(t,\epsilon)= \epsilon\bigl[(\cos\nu t-\cos\zeta t)M+\sin\nu t\,N\bigr]

(4.3)

vanishes at both endpoints. Fourier orthogonality makes its first endpoint derivative zero. Add a linear combination of the finite controls from lemma 4.1,

D(t,\epsilon)=D_{\mathrm{tr}}(t,\epsilon) +\sum_{\ell=1}^{d(2d+1)}c_\ell(\epsilon)D_\ell(t).

(4.4)

The ordinary analytic implicit-function theorem, in a chart at $I\in\Sp(2d,\R)$ , gives $c_\ell(\epsilon)=O(\epsilon^2)$ and $\mathsf S(2\pi)=I$ exactly. For sufficiently small $\epsilon$ , $K_0+D(t,\epsilon)>0$ . Thus these are finite-duration exact quantum returns with local controls and fixed off-block entries. Only a finite-dimensional endpoint equation has been inverted.

4.2 The actual Gaussian configuration curvature

Starting in $\psi_0$ , a quadratic history remains Gaussian:

\psi_t(q)=c_t\exp[-q^TA(t)q/2+iq^TB(t)q/2], \qquad A(t)>0,\quad B(t)=B(t)^T .

Substitution in the Schrödinger equation gives

\dot A=-AB-BA,\qquad \dot B=A^2-B^2-K,\qquad \dot L=BL,\quad L(0)=I .

(4.5)

The last equation is the actual guided map $q(t)=L(t)q(0)$ . To see that these formulas are global for finite time, write

X=\mathsf S_{qq}+i\mathsf S_{qp}A_0,\qquad Y=\mathsf S_{pq}+i\mathsf S_{pp}A_0 .

Symplecticity gives $X^*Y-Y^*X=2iA_0$ . Hence $X$ is invertible, $A-iB=-iYX^{-1}$ is symmetric, and $A=X^{-*}A_0X^{-1}>0$ . The Gaussian has no nodes, $B$ is bounded on every finite time interval, and $L$ is a global linear diffeomorphism. Also

L(t)^TA(t)L(t)=A_0,\qquad \det L(t)=\exp\left(\int_0^t\tr B(s)\,ds\right)>0 .

(4.6)

Every Gaussian ray return therefore gives $A_0^{1/2}L(T)A_0^{-1/2}\in SO(d)$ .

Proposition 4.2 (Curvature surviving exact repair)

For the corrected loops (4.4), in modal coordinates, $L(T)=I+\epsilon^2L_2+O(\epsilon^3)$ and

\begin{align}\skw L_2 &=-\pi\nu[\mathcal R_\nu\widetilde M, \mathcal R_\nu\widetilde N], &(\mathcal R_\nu Q)_{ij} &=\frac{Q_{ij}}{(\omega_i+\omega_j)^2-\nu^2},\tag{4.7}\\ G_{ij} &=\frac{2\sqrt{\omega_i\omega_j}}{\omega_i+\omega_j} (\skw L_2)_{ij}, &\Omega^{1/2}L(T)\Omega^{-1/2} &=I+\epsilon^2G+O(\epsilon^3). \tag{4.8}\end{align}

Matrix commutators in (4.7) have the convention $[P,Q]=PQ-QP$ .

Proof

Let $A=\Omega+\epsilon a+\cdots$ and $B=\epsilon b+\cdots$ . For $\sigma_{ij}=\omega_i+\omega_j$ the linearized equations are $\dot a_{ij}=-\sigma_{ij}b_{ij}$ and $\dot b_{ij}=\sigma_{ij}a_{ij}-D_{ij}$ . A forcing $\widetilde M_{ij}\cos\nu t$ contributes

b_{ij}(t)=\widetilde M_{ij} \frac{\nu\sin\nu t-\sigma_{ij}\sin\sigma_{ij}t} {\sigma_{ij}^2-\nu^2};

a forcing $\widetilde N_{ij}\sin\nu t$ contributes

b_{ij}(t)=\widetilde N_{ij} \frac{\nu(\cos\sigma_{ij}t-\cos\nu t)} {\sigma_{ij}^2-\nu^2}.

There is an analogous cosine expression at $\zeta$ . All these functions integrate to zero over $T=2\pi$ . Expansion of $\dot L=BL$ yields

\skw L_2=\frac12\int_0^{2\pi} \left[b(t),\int_0^t b(s)\,ds\right]dt .

(4.9)

The direct integral of the second-order $B$ is symmetric, including the contribution of the endpoint repair.

Orthogonality removes unequal frequencies from (4.9). At a natural frequency $\sigma_{ij}$ all matrices are multiples of the same symmetric matrix unit, and their commutator is zero. The only nonzero paired sine and cosine matrices are at $\nu$ : they are $\nu\mathcal R_\nu\widetilde M$ and $-\nu\mathcal R_\nu\widetilde N$ . Their elementary integral is $-\pi\nu[\mathcal R_\nu\widetilde M, \mathcal R_\nu\widetilde N]$ , proving (4.7).

Exact return and (4.6) imply $\Omega L_2+L_2^T\Omega=0$ . Solving this entrywise in terms of $\skw L_2$ gives (4.8). This is why a symmetric contribution before whitening cannot be simply ignored after whitening: the covariance identity supplies the necessary relation.

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4.3 From return curves to the exact group

Lemma 4.3 (Exact group generation)

Suppose a subgroup $\mathscr G\subset SO(d)$ contains analytic curves $R_j(\epsilon)=I+\epsilon^2G_j+O(\epsilon^3)$ and their inverses. If the $G_j$ generate $\mathfrak{so}(d)$ as a Lie algebra, then $\mathscr G=SO(d)$ .

Proof

For a sufficiently small fixed nonzero $a$ , the curves $C_j(t)=R_j(a+t)R_j(a)^{-1}$ pass through the identity and have velocities $X_j=2aG_j+O(a^2)$ . A finite list of Lie words in the $G_j$ is a basis, so the corresponding determinant remains nonzero for $X_j/(2a)$ . Let $V$ be the span of $\operatorname{Ad}_hX_j$ for all $h$ in the subgroup generated by the $C_j$ . It is invariant under these adjoint actions. Differentiating $\operatorname{Ad}_{C_j(t)}V=V$ gives $[X_j,V]\subset V$ . Thus $V=\mathfrak{so}(d)$ . Select finitely many adjoint velocities forming a basis. The product of their conjugated curves has surjective differential at the origin, so its image contains an identity neighborhood, by the finite-dimensional inverse-function theorem. Consequently $\mathscr G$ is open. Since $SO(d)$ is connected, $\mathscr G=SO(d)$ . Every element obtained this way is a finite product of actual curves and their inverses, rather than only a limit of such products.

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