Caustic-time Mehler formulas for inhomogeneous quadratic Hamiltonians and the exponential image of the metaplectic semigroup (Hörmander 1995)

Hörmander's 1995 paper on the symplectic classification of quadratic forms determined the Weyl symbol of the exponential of every Weyl-quantized quadratic form: away from the caustic times, where the non-degeneracy determinant does not vanish, the symbol is the Gaussian given by the general Mehler formula, and at the caustic times of the homogeneous case the paper itself describes the symbol as an explicit singular Gaussian, a Gaussian times the Lebesgue measure on the range of the relevant flow matrix. Two regimes were left untreated. First, for inhomogeneous quadratic polynomials, whose exponentials govern Stark-type and other linearly perturbed equations, Theorem 4.7 of that paper computes the symbol only under a non-degeneracy hypothesis and says nothing at the caustic times. Second, at the semigroup level, where the paper proves that the operators exp(iQ^w) with Im Q at least zero generate the metaplectic semigroup (the double cover of the semigroup of positive complex symplectic matrices), the range of the exponential map Q mapped to exp(iQ^w) itself is not characterized. The later literature treats these regimes by avoidance or explicit postponement: time-dependent Mehler formulas retain non-caustic hypotheses (Pravda-Starov 2018), the limiting degree-2 case is explicitly postponed (Viola), inhomogeneous semigroup extensions are deferred (Alphonse-Bernier), and the 2026 structural theory of the metaplectic semigroup states that Mehler formulas hold only outside the caustics. The problem asks for both (A) an explicit caustic-time Mehler formula for the Weyl symbol of exp(itQ^w) for inhomogeneous quadratic polynomials Q, and (B) an intrinsic characterization of the exponential image inside the metaplectic semigroup.

Background

Phase space is $\mathbb{R}^{2n}=T^*\mathbb{R}^n$ with points $X=(x,\xi)$ and the canonical symplectic form $\sigma((x,\xi),(y,\eta))=\langle \xi,y\rangle-\langle x,\eta\rangle$, the non-degenerate antisymmetric bilinear form that pairs momentum with displacement. To a symbol $a(x,\xi)$ one associates its Weyl quantization $a^w(x,D)$, the standard phase-symmetric pseudodifferential operator that places the $x$- and $\xi$-factors symmetrically in each monomial; when $a$ is a polynomial of degree at most 2, $a^w(x,D)$ is a differential operator with coefficients of degree at most 2 in $(x,D_x)$. For a quadratic form $q$ with polarized symmetric bilinear form, the Hamilton map $F$ is the unique linear map on phase space satisfying $\sigma(Y,FX)=q(Y,X)$ for all $X,Y$; equivalently, if $q(X)=\langle QX,X\rangle$ with $Q$ symmetric, then $F=\sigma Q$. The map $F$ encodes the form in symplectic terms, and all the formulas below are expressed through it.

The classical formula of Mehler for the harmonic oscillator kernel was generalized by Hörmander (1995) to every quadratic form: under the sign condition $\operatorname{Re} q \le 0$, for $t\ge 0$ at which $\det \cos(tF)\ne 0$, the Weyl symbol of the heat-type evolution operator $\exp(tq^w)$ is the Gaussian

$$(\det(\cos(tF)))^{-1/2} \exp(\sigma(X, \tan(tF)X)),$$

and the Schrödinger-type operator $\exp(itq^w)$ is covered by the corresponding analytic substitution in $t$. For the harmonic-oscillator Schrödinger flow, whose symbol is $(\cos t)^{-1} \exp(-i(x^2+\xi^2) \tan t)$ away from the singular times, the paper shows that at $t=\pi/2+k\pi$ the symbol is the Dirac mass $i(-1)^{k+1}\pi \delta_0$ (quoted with page reference by Pravda-Starov, 2018). The times at which the non-degeneracy determinant vanishes are the caustic times, the exact-counterpart of the caustics of geometric optics: there the prefactor blows up and the symbol is no longer a function but a distribution. In the homogeneous case the paper gives a complete caustic-time answer: the Weyl symbol is a Gaussian times the Lebesgue measure on the range of $\cos(tF)$, described through a symplectic form $\sigma_F$ induced on that range and a quadratic form $E_F$ (the zbMATH review of the paper summarizes the symbol as always a Gaussian multiplied by a Lebesgue measure in the range of $\cos(tF)$). The harmonic-oscillator Dirac mass is the degenerate special case where that range collapses to the origin.

An inhomogeneous quadratic polynomial $Q(X)=q(X)+2\sigma(X,a)+b$, with $a\in \mathbb{R}^{2n}$ and constant $b\in \mathbb{C}$, adds a linear and a constant term; its Weyl quantization covers, for example, Schrödinger operators in a constant external field (Stark-type operators, whose classical explicit propagator goes back to Avron and Herbst) and translated or linearly forced oscillators. Theorem 4.7 of the 1995 paper (its formula (4.21)) computes the Weyl symbol of $\exp(itQ^w)$ for such $Q$, under the sign condition that the imaginary part of $Q$ is bounded below, at all times satisfying the non-degeneracy hypothesis $\det \cosh(tF)\ne 0$ (in the normalization of Theorem 4.7; in equivalent normalizations of later papers the condition reads $\det \cos(tF)\ne 0$ after analytic continuation in $t$). The symbol is the homogeneous Gaussian factor $(\det \cosh(tF))^{-1/2} \exp(i\sigma(X, \tanh(tF)X))$ times an explicit affine-phase exponential $\exp(i(2\sigma(X, F^{-1} \tanh(tF)a) - \sigma(a, F^{-2}(\tanh(tF)-tF)a) + tb))$. Unlike the homogeneous case, the caustic times of the inhomogeneous flow, at which this hypothesis fails, are not analyzed in the paper: no formula is given there.

The second half of the paper passes from single exponentials to semigroups. The metaplectic group $Mp(n,\mathbb{R})$ is the double cover of the real symplectic group $Sp(2n,\mathbb{R})$, realized on $L^2(\mathbb{R}^n)$ by unitary operators with pure-phase Gaussian kernels, such as the Fourier transform. In the complexified picture, a complex symplectic matrix $S\in Sp(2n,\mathbb{C})$ is called positive when $i^{-1}(\sigma(Sz,Sz)-\sigma(z,z))\ge 0$ for all $z\in \mathbb{C}^{2n}$, with $\sigma$ complexified; the positive matrices form a semigroup, denoted $C_+$ by Hörmander and $Sp^+(2n,\mathbb{C})$ in the later literature. Above it sits the metaplectic semigroup, Hörmander's version consisting of operators on $L^2(\mathbb{R}^n)$ with Gaussian kernels forming a double cover of $C_+$; equivalent incarnations are Brunet's metaplectic semigroup and Howe's oscillator semigroup on the Bargmann-Fock space of entire functions (Hilgert proved the two theories isomorphic, and Giacchi-Rodino-Tramontana 2026 give a modern account). The bridge between the two halves of the paper is that $\exp(iQ^w)$, for a quadratic form with $\operatorname{Im} Q \ge 0$, is such a Gaussian-kernel operator (in the paper's notation, $\exp(iQ^w)$ corresponds to the positive map $\exp(-2F)$, up to the standard factors of 2 and $i$ that vary across the literature), and the semigroup generated by all these exponentials is the metaplectic semigroup. What the paper does not provide is a characterization of the range of the exponential map itself: which elements of the metaplectic semigroup, or equivalently which positive symplectic maps, are single exponentials $\exp(iQ^w)$ with $\operatorname{Im} Q \ge 0$. A remark at the end of Section 5 observes that this map is not locally surjective, and no description of its image is given.

This calculus is the exact-computation backbone of a large literature: propagators of heat and Schrödinger equations with quadratic Hamiltonians underlie observability and null-controllability results (steering solutions to zero by a control), propagation of phase-space (Gabor) singularities, and exact operator-splitting schemes, while the metaplectic semigroup underlies time-frequency analysis and representation theory. The exact formulas, however, are available away from the caustics only in the inhomogeneous case, and the semigroup structure theory describes generated subsemigroups, not the exponential image.

Problem Statement

The problem is to complete Hörmander's general Mehler-formula framework for the exponential of a Weyl-quantized quadratic polynomial, by closing the two regimes that the 1995 paper left untreated. The common objective is a complete and explicit description of the exponential map $Q \mapsto \exp(iQ^w)$ on quadratic polynomials with imaginary part bounded below: its output at every time (component A) and its image as a semigroup element (component B).

(A) Caustic-time inhomogeneous Mehler formula. For an inhomogeneous quadratic polynomial $Q(X)=q(X)+2\sigma(X,a)+b$ with imaginary part bounded below, give an explicit formula for the Weyl symbol of $\exp(itQ^w)$ at the caustic times, that is, at the times excluded by the non-degeneracy hypothesis of Theorem 4.7 (where $\det \cosh(tF)=0$ in that theorem's normalization). The answer is an explicitly computable tempered distribution on $\mathbb{R}^{2n}$. The formula must be proved to be the limit, in the space $\mathscr{S}'(\mathbb{R}^{2n})$ of tempered distributions, of the known non-caustic symbol as $t$ approaches the caustic time through regular times, and it must specialize to the homogeneous caustic formula of the paper when the linear and constant terms vanish, reproducing in particular the harmonic-oscillator Dirac mass.

(B) Exponential image of the metaplectic semigroup. Characterize, by an intrinsic criterion decidable from the element itself, the range of the map $Q \mapsto \exp(iQ^w)$ from quadratic forms with $\operatorname{Im} Q \ge 0$ into the metaplectic semigroup (equivalently, in the normalization of the source, of the map $F \mapsto \exp(-2F)$ on Hamilton maps of such forms into the semigroup of positive symplectic matrices): determine which positive symplectic maps, or their metaplectic lifts, arise as single exponentials of Hamilton maps of forms with non-negative imaginary part. The criterion must not presuppose a choice of matrix-logarithm branch; sufficiency and necessity must be proved; and the criterion must be consistent with the known elements of the image, such as the elements arising from real symplectic flows and the Gaussian contractions and rotations that generate the semigroup, as well as with elements known or provably not of this form.

The two components are separable in principle, but they arise together as the two unanalyzed regimes of the same source framework and are stated here as one problem with that common objective.

References

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