Determining metrics from the scattering map of the time-dependent Schrödinger equation

This paper establishes that for a specific class of time-dependent metrics, the scattering maps of two Schrödinger operators differ by a compact operator if and only if the metrics are related by a diffeomorphism pull-back.

The Gist

Imagine you are standing in a vast, empty field. Suddenly, a storm rolls in. This storm isn't just wind and rain; it's a shifting, invisible landscape that bends the path of anything moving through it. You can't see the storm itself, but you can watch how a ball thrown into the field behaves.

If you throw the ball from the left, it might curve, speed up, or slow down as it passes through the storm, and then exit to the right. By watching exactly how the ball comes out (its speed, direction, and timing), can you figure out exactly what the storm looked like inside?

This is the core question of the paper "Determining Metrics from the Scattering Map of the Time-Dependent Schrödinger Equation" by Qiuye Jia.

Here is a breakdown of the paper's ideas using simple analogies:

1. The Setup: The "Time-Traveling" Storm

In physics, the Schrödinger equation describes how tiny particles (like electrons) move. Usually, we assume the space they move through is flat and unchanging (like a calm, flat field).

But in this paper, the author considers a more complex scenario:

• The Metric: Think of this as the "shape" of the space. It could be a curved valley or a bumpy hill.
• Time-Dependent: This shape isn't static. It changes as time passes. Maybe the hill rises and falls, or the valley shifts position.
• The Scattering Map: This is the "fingerprint" of the storm. It's a mathematical rule that says: "If a particle enters the storm with this specific speed and direction, it will leave with this specific speed and direction."

2. The Problem: The "Black Box" Mystery

The author asks: If we only know the "fingerprint" (the scattering map), can we reconstruct the "storm" (the metric)?

There is a catch. Imagine two different storms:

• Storm A: A real, physical hill that bends the path.
• Storm B: A flat field, but you are wearing special glasses that make the flat field look like a hill.

If you just look at the path of the ball, Storm A and Storm B might look identical. In math terms, they are related by a diffeomorphism (a fancy word for a smooth stretching or reshaping of space that doesn't tear it).

The paper proves a "Yes, but..." answer:

If two time-varying landscapes produce the exact same scattering map (up to a tiny, negligible difference), then the landscapes must be the same, except that one might just be a reshaped version of the other.

3. The Solution: How the Author Solved It

The author didn't just guess; they used a very sophisticated toolkit called Microlocal Analysis. Here is how that works in our analogy:

The "Flashlight" Technique (Microlocalization)

Imagine trying to see a hidden object in a dark room. You can't just turn on a light; you need a laser pointer that can focus on incredibly tiny details.

• The author uses a mathematical "laser" to zoom in on the scattering map.
• They don't just look at where the ball goes; they look at the phase (the timing and rhythm of the wave) of the particle as it exits.

The "Echo" and the "Sojourn Time"

When a particle travels through the storm, it spends a certain amount of time inside.

• The Analogy: Imagine shouting into a canyon. The time it takes for the echo to return tells you how far the walls are.
• The Math: The author shows that the scattering map contains a hidden "echo" called the sojourn time. This is the exact amount of time the particle spent traveling through the curved space compared to how long it would have taken in a flat space.
• By measuring this "echo" with extreme precision, they can calculate the length of the geodesics (the shortest paths) inside the storm.

The "Lens" of Geometry

Once they know the length of every possible path through the storm, they have what mathematicians call Lens Data.

• Think of this like a CT scan. If you know how long X-rays take to travel through every part of a body, you can reconstruct the body's shape.
• The author proves that if you have the Lens Data (path lengths and entry/exit points), you can mathematically reconstruct the shape of the space (the metric), provided the space has certain "convex" properties (like a bowl shape rather than a saddle shape).

4. The "Secret Sauce": Second Microlocalization

This is the most creative part of the paper.
Usually, when you look at a wave, you see the big waves. But sometimes, the important information is hidden in the tiny ripples on top of the big waves.

• The author invented a way to "blow up" the mathematical view to see these tiny ripples.
• They call this Second Microlocalization.
• The Metaphor: Imagine listening to a symphony. Most people hear the melody. The author built a device that isolates the specific vibration of the violin string while the orchestra is playing, allowing them to hear the exact tension of the string. This extra layer of detail is what allowed them to distinguish between two landscapes that looked identical at a lower level of detail.

Summary

The Big Picture:
The paper solves a puzzle about time-traveling shapes. It proves that if you watch how particles scatter through a changing, curved universe, you can figure out exactly what that universe looks like (up to a reshaping).

Why it matters:
This is a huge step in Inverse Problems. In the real world, we often can't see the inside of things (like the Earth's core, a black hole, or a human body). We can only measure how waves (sound, light, seismic waves) bounce off them. This paper gives mathematicians a new, powerful set of tools to reconstruct the hidden geometry of the universe from those bounces, even when the universe is changing as we watch it.

In one sentence:
By analyzing the precise "echoes" of particles bouncing off a shifting, curved space, the author proved we can mathematically reconstruct the shape of that space, provided we look closely enough at the timing of the waves.

Technical Summary

1. Problem Statement

The paper addresses an inverse problem for the time-dependent Schrödinger equation on $\mathbb{R}^{n+1}$ (where $z \in \mathbb{R}^n$ and $t \in \mathbb{R}$). The equation is given by:
$$P u = (D_t + \Delta_{g(t)} + V(z,t))u = 0$$
where:

• $D_t = -i\partial_t$.
• $\Delta_{g(t)}$ is the Laplace-Beltrami operator associated with a smooth, time-dependent Riemannian metric $g(t)$.
• $V(z,t)$ is a smooth, complex-valued potential.
• Both the metric perturbation $\tilde{g} = g - g_0$ and the potential $V$ are compactly supported in spacetime (i.e., they vanish outside a large ball $B_R(0)$ and time interval $[-T, T]$).
• The metric is assumed to be non-trapping (geodesics escape any compact set in finite time).

The Scattering Map:
For global solutions $u$, the scattering map $S$ relates the asymptotic profile of the solution as $t \to -\infty$ (incoming data $f_-$) to the profile as $t \to +\infty$ (outgoing data $f_+$).
$$S: f_- \mapsto f_+$$
The map is known to be a Fourier Integral Operator (FIO) of a specific type (Legendre distribution).

The Core Question:
If two such Schrödinger operators $P_1$ and $P_2$ (with metrics $g_1, g_2$ and potentials $V_1, V_2$) have scattering maps $S_{g_1}$ and $S_{g_2}$ that differ only by a compact operator on $L^2(\mathbb{R}^n)$, does this imply that the metrics are related by a diffeomorphism? Specifically, does there exist a diffeomorphism $\psi: \mathbb{R}^{n+1} \to \mathbb{R}^{n+1}$ preserving time slices (i.e., $\psi(t, z) = (t, \psi_1(t, z))$) such that $g_1 = \psi^* g_2$?

2. Methodology

The proof relies on a sophisticated synthesis of geometric inverse problems and microlocal analysis, specifically utilizing advanced pseudodifferential calculi adapted to the geometry of the problem.

A. Microlocal Frameworks

The author employs three distinct but interconnected pseudodifferential algebras to analyze the singularities of the scattering map at different scales:

1. Scattering Calculus ($sc$): Used to analyze behavior at spatial infinity ($|z| \to \infty$).
2. Parabolic Scattering Calculus ($ps$): Used for the time-dependent Schrödinger operator, where the fiber infinity is compactified parabolically (reflecting the $t \sim |z|^2$ scaling of the Schrödinger equation).
3. 1-Cusp Calculus ($1c$): Used to analyze the boundary behavior and the "1-cusp" geometry arising from the interaction between the bulk and the boundary at infinity.

B. The 1c-ps and 1c-1c Geometry

• Bulk-Boundary Duality: The paper establishes a duality between the parabolic scattering cotangent bundle over the spacetime bulk ($psT^*\mathbb{R}^{n+1}$) and the 1-cusp cotangent bundle over the boundary ($1cT^*\mathbb{R}^n$).
• Scattering Map Structure: The scattering map $S$ is characterized as an elliptic 1c-1c Fourier Integral Operator of order 0. Its canonical relation is the graph of the classical scattering map ($Cl_g$), which maps incoming bicharacteristics to outgoing ones.
• Second Microlocalization: To extract geometric information (specifically the length of geodesics) hidden in the subleading terms of the phase function, the author introduces second microlocalization. This involves blowing up the phase space at fixed radial frequencies to create a new "scattering" level of frequencies. This allows the recovery of the sojourn time (the time a geodesic spends in the perturbed region) from the analytic data of the scattering map.

C. Geometric Inverse Problems

The analytic results are linked to the Lens Rigidity Problem. The paper demonstrates that the scattering map encodes:

1. The Truncated Scattering Relation: The map sending the entry point and direction of a geodesic into a large ball $B_R(0)$ to its exit point and direction.
2. The Sojourn Time (Lens Data): The total time a geodesic spends inside the perturbation region.

The proof concludes by invoking a rigidity theorem by Stefanov, Uhlmann, and Vasy, which states that if two metrics have the same lens data (scattering relation and length) and one admits a convex function (a condition related to curvature, e.g., non-positive or non-negative sectional curvature), then the metrics are diffeomorphic via a boundary-fixing map.

3. Key Contributions

1. First Result for Time-Dependent Metrics: This is the first result establishing the determination of a time-dependent metric from the scattering map of the time-dependent Schrödinger equation. Previous results were largely restricted to time-independent settings or wave equations.
2. Characterization of the Scattering Map: The paper rigorously characterizes the scattering map as a 1c-1c Fourier Integral Operator and derives its principal symbol and composition laws.
3. Recovery of Lens Data from Analytic Data:
   • It proves that the scattering relation (geometric entry/exit data) is encoded in the restriction of the classical scattering map to the boundary of the 1-cusp phase space.
   • It proves that the geodesic length (sojourn time) is encoded in the 1-jet (specifically the value of the phase function at critical points) of the lifted graph of the scattering map.
4. Second Microlocalization Technique: The author develops a technique to "blow up" the phase space to separate different 1-jets of Lagrangian submanifolds that share the same boundary. This allows the extraction of the sojourn time, which is otherwise invisible in the leading-order principal symbol.
5. Rigidity Theorem: The main theorem (Theorem 1.3) establishes that if $S_{g_1} - S_{g_2}$ is compact, then $g_1 = \psi^* g_2$, provided one of the metrics admits a convex function.

4. Main Results

Theorem 1.3 (Main Theorem):
Let $g_1(t)$ and $g_2(t)$ be two time-dependent metrics satisfying the compact support and non-trapping conditions. Suppose that either $g_1$ or $g_2$ admits a convex function (in the sense of Definition 1.1, e.g., satisfying specific curvature bounds).
If the scattering maps satisfy:
$$S_{g_1} - S_{g_2} \in \mathcal{K}(L^2(\mathbb{R}^n))$$
(i.e., their difference is a compact operator), then there exists a diffeomorphism $\psi: \mathbb{R}^{n+1} \to \mathbb{R}^{n+1}$ of the form $\psi(t, z) = (t, \psi_1(t, z))$ such that:
$$g_1 = \psi^* g_2$$
Furthermore, this implies that the other metric also admits a convex function.

Key Intermediate Results:

• Proposition 9.1: The classical scattering map at infinity determines the truncated scattering relation inside the compact support region.
• Theorem 9.3: If two metrics are "microlocally lens equivalent" (their lifted scattering graphs have the same boundary and same $N_1$ component encoding sojourn time), then their geodesic lengths are identical.
• Proposition 9.6: The condition that $S_{g_1} - S_{g_2}$ is compact implies that the two metrics are microlocally lens equivalent.

5. Significance

• Theoretical Advancement: The paper bridges the gap between the microlocal analysis of time-dependent PDEs (specifically the Schrödinger equation) and geometric inverse problems. It extends the scope of scattering theory from static to dynamic geometries.
• Methodological Innovation: The use of 1-cusp and second microlocalization techniques provides a powerful new toolkit for extracting geometric invariants (like travel time) from the subleading terms of oscillatory integrals. This is crucial because the leading order of the scattering map often only sees the "direction" of rays, while the "time" (length) is hidden in the phase.
• Physical Implications: The result suggests that in quantum systems evolving in time-varying curved spaces (or inhomogeneous media), the scattering data at infinity contains complete information about the internal geometry of the medium, up to a coordinate transformation.
• Future Directions: The author notes that while the metric is determined, the potential $V$ is not detected at this order (as it affects lower-order symbols). The framework sets the stage for determining potentials and higher-order geometric invariants by analyzing lower-order terms of the scattering map.

In summary, Jia's work provides a rigorous proof that the scattering map of a time-dependent Schrödinger equation uniquely determines the underlying time-dependent metric (up to a diffeomorphism), utilizing a novel blend of geometric analysis and advanced microlocal calculus.