Classical Ghost Scattering Map

• The Classical Ghost Scattering Map is a symplectic transformation on T*(S^(d-1)) that quantizes the scattering process of semiclassical Gaussian states.
• It underpins the semiclassical scattering framework by linking quantum scattering operators to classical trajectory deflections and phase-space transport in compactly supported potentials.
• The map is realized via coherent state propagation, encoding key dynamics such as action phase shifts, width updates, and linearized Hamiltonian flow corrections.

In semiclassical scattering theory for compactly supported, non-trapping perturbations of the Laplacian on $\mathbb{R}^d$ at fixed energy, the Classical Ghost Scattering Map is the classical scattering map

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$

acting on phase-space points $\rho=(\omega,\eta)$, where the corresponding “ghosts” are semiclassical Gaussian, or coherent, states on $\mathbb{S}^{d-1}$ microlocally concentrated near $\rho$. In Ingremeau’s formulation, the semiclassical scattering matrix $S_h$ transports such a Gaussian state to another Gaussian state centered at $\kappa(\rho)$, with an additional action phase and a width/amplitude update determined by the linearized Hamiltonian flow. In this sense, the scattering matrix quantizes the classical scattering map (Ingremeau, 2016).

1. Semiclassical scattering framework

The setting is the semiclassical regime $h\to 0$ for compactly supported, non-trapping perturbations of the Laplacian on $\mathbb{R}^d$, with fixed energy $E=\tfrac12$. The operator is

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$0

where $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$1, and the relevant energy layer is

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$2

The non-trapping hypothesis requires that for every $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$3 there exists $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$4 such that

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$5

where $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$6 is the Hamiltonian flow of $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$7 and $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$8 is the base projection (Ingremeau, 2016).

At energy $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$9, for each smooth incoming datum $\rho=(\omega,\eta)$0, there is a unique solution of

$\rho=(\omega,\eta)$1

with asymptotics

$\rho=(\omega,\eta)$2

The semiclassical scattering matrix is then defined by

$\rho=(\omega,\eta)$3

It extends by density to a unitary operator on $\rho=(\omega,\eta)$4, and $\rho=(\omega,\eta)$5 is trace class. The phase prefactor is chosen so that $\rho=(\omega,\eta)$6 when $\rho=(\omega,\eta)$7 (Ingremeau, 2016).

This framework isolates the fixed-energy scattering information in an operator on the sphere of asymptotic directions. The “ghost” construction turns that operator into a microlocal transport law on $\rho=(\omega,\eta)$8.

2. Classical scattering map on $\rho=(\omega,\eta)$9

The cotangent bundle is identified as

$\mathbb{S}^{d-1}$0

Given an incoming direction $\mathbb{S}^{d-1}$1 and impact parameter $\mathbb{S}^{d-1}$2, choose $\mathbb{S}^{d-1}$3 with $\mathbb{S}^{d-1}$4. There exists a unique phase-space point $\mathbb{S}^{d-1}$5 whose base trajectory agrees for large negative times with the free incoming line

$\mathbb{S}^{d-1}$6

By non-trapping, for large positive times the trajectory is again free: $\mathbb{S}^{d-1}$7 for some $\mathbb{S}^{d-1}$8, $\mathbb{S}^{d-1}$9, and $\rho$0. The classical scattering map is

$\rho$1

Equivalently, if for the asymptotic free motions one writes

$\rho$2

then the scattering relation is

$\rho$3

under the identification $\rho$4 (Ingremeau, 2016).

Microlocally on the non-trapped subset, $\rho$5 is a canonical, hence symplectic, transformation. Geometrically, it records the deflection of an incoming free line $\rho$6 into an outgoing free line with new direction $\rho$7 and transverse offset $\rho$8. The object quantized by $\rho$9 is therefore not merely the change of direction, but the full canonical relation on asymptotic phase space.

3. Gaussian ghosts on the sphere

The “ghosts” are semiclassical Gaussian states on $S_h$0 concentrated near a point of $S_h$1. In geodesic normal coordinates $S_h$2 centered at $S_h$3, a normalized Gaussian wave packet centered at $S_h$4 is

$S_h$5

Its spatial mass is centered at $S_h$6, hence at $S_h$7 on the sphere, while its tangential Fourier content is centered at $S_h$8 (Ingremeau, 2016).

The paper also uses a broader family of Gaussian states

$S_h$9

depending on $\kappa(\rho)$0, a symmetric matrix $\kappa(\rho)$1 with $\kappa(\rho)$2, a polynomial $\kappa(\rho)$3, and a cutoff $\kappa(\rho)$4. The associated phase-space point on the sphere is

$\kappa(\rho)$5

Removing the cutoff produces a slightly simpler model differing by $\kappa(\rho)$6, so the two are interchangeable microlocally (Ingremeau, 2016).

A crucial structural fact is a resolution of identity by these coherent states. Writing $\kappa(\rho)$7, one has for $\kappa(\rho)$8

$\kappa(\rho)$9

with

$h\to 0$0

This validates Gaussian ghosts as a complete microlocal testing family on $h\to 0$1: they are not merely illustrative packets, but a coherent-state frame capable of probing the full action of the scattering matrix (Ingremeau, 2016).

4. Transport theorem: $h\to 0$2 moves ghosts by $h\to 0$3

The central result states that the scattering matrix preserves the Gaussian character of these states while moving their centers by the classical scattering map. For $h\to 0$4, symmetric $h\to 0$5 with $h\to 0$6, and polynomial $h\to 0$7, there exist $h\to 0$8, a phase $h\to 0$9, a symmetric $\mathbb{R}^d$0 with $\mathbb{R}^d$1, and polynomials $\mathbb{R}^d$2 such that for each $\mathbb{R}^d$3,

$\mathbb{R}^d$4

with

$\mathbb{R}^d$5

If

$\mathbb{R}^d$6

then

$\mathbb{R}^d$7

Thus the outgoing packet is centered exactly at the classical image of the incoming phase-space point (Ingremeau, 2016).

The phase $\mathbb{R}^d$8 is a classical action: $\mathbb{R}^d$9 where

$E=\tfrac12$0

with $E=\tfrac12$1 and $E=\tfrac12$2 chosen so that $E=\tfrac12$3 is outgoing. The Gaussian width evolves through the linearized Hamiltonian flow: if

$E=\tfrac12$4

then

$E=\tfrac12$5

At leading order the amplitude polynomial has principal coefficient

$E=\tfrac12$6

and $E=\tfrac12$7 has degree at most $E=\tfrac12$8 (Ingremeau, 2016).

A simplified microlocal summary is

$E=\tfrac12$9

for some $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$00, where $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$01 is the classical action along the scattering trajectory issued from $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$02, and $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$03 is an $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$04-asymptotic amplitude determined by the linearized Hamiltonian flow. This summarizes the exact theorem without replacing it (Ingremeau, 2016).

5. Fourier integral operator interpretation and microlocality

The coherent-state theorem complements an existing Fourier integral operator description. Alexandrova’s result, as cited in the paper, shows that microlocally near non-trapped points the scattering matrix $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$05 is a semiclassical Fourier integral operator associated with the canonical transformation $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$06 on $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$07. In the standard FIO sense, $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$08 therefore quantizes $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$09 (Ingremeau, 2016).

The Gaussian-state approach makes that quantization explicit at the level of individual packets. It exhibits the transported center, the action phase, the propagated width, the amplitude expansion, and the error estimate. It also shows a strong microlocal locality property: $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$10, $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$11, and the polynomials $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$12 depend only on $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$13, $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$14, and on $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$15 in an arbitrarily small neighborhood of the single classical scattering trajectory. This suggests a particularly sharp form of trajectorywise microlocal control, stronger in presentation than a purely canonical-relation statement (Ingremeau, 2016).

Because the Gaussian states resolve the identity on $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$16, the coherent-state transport law is not restricted to special inputs. It provides a microlocal probe of the entire scattering matrix: feeding in packets centered at $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$17 and reading off the center of the outgoing packet recovers $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$18, while the phase and amplitude encode the linearized dynamics and action along the underlying classical orbit.

6. Hypotheses, geometric special cases, and terminological distinctions

The theory assumes $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$19, compact support, fixed energy $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$20, non-trapping on $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$21, and dimension $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$22. Under these hypotheses the outgoing resolvent satisfies the non-trapping bound

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$23

which is used in the construction of generalized eigenfunctions and in the error analysis. In dimension $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$24, the error bookkeeping requires minor adjustments; specifically, one may increase $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$25 by $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$26 to retain the stated rate (Ingremeau, 2016).

For radial potentials, no explicit formula for $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$27 is given in the paper, but the expected structure is stated: the impact parameter $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$28 is conserved, $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$29 is rotated by a deflection angle depending on $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$30 and the energy, and $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$31 is determined by that deflection angle together with the transverse shift. The Gaussian-state theorem applies directly to such cases (Ingremeau, 2016).

The term ghost requires disambiguation. In the semiclassical scattering context above, it does not refer to gauge-theoretic Faddeev–Popov ghosts; it denotes microlocal Gaussian states on $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$32 concentrated near points of $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$33 (Ingremeau, 2016). In a different literature, classical ghost imaging uses “ghost” for an imaging modality reconstructed from correlations between illumination patterns and bucket measurements. There the relevant operator is a correlation map

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$34

and under whitened statistics it recovers a transport density $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$35 up to scale, linking classical ghost imaging to dual photography via reciprocity (Sen, 2013). In random waveguides, ghost imaging is again formulated through a cross-correlation functional

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$36

which remains effective even in the equipartition regime when conventional imaging fails (Borcea et al., 2018). A further, again distinct, usage occurs in long-range scattering, where the scattering matrix is a Fourier integral operator with phase generated by a modified classical scattering map $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$37 incorporating long-range action corrections through a generating function $\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$38 (Nakamura, 2018).

Within semiclassical scattering proper, however, the Classical Ghost Scattering Map is precisely the classical map

$\kappa:T^*\mathbb{S}^{d-1}\to T^*\mathbb{S}^{d-1}$39

read off from the transport of Gaussian ghosts by the fixed-energy scattering matrix. Its significance lies in giving a concrete, packet-level realization of the principle that the quantum scattering operator is the quantization of the classical scattering dynamics.