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arxiv: 2606.26581 · v1 · pith:USXBDDMDnew · submitted 2026-06-25 · 🧮 math.AP · math-ph· math.MP

Determining potentials from the scattering map of the time-dependent Schr\"odinger equation

Qiuye Jia This is my paper

Pith reviewed 2026-06-26 04:20 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP
keywords scattering maptime-dependent Schrödinger equationinverse scattering problempotentialsPoisson operatorscompact operatorscurved spaceRiemannian metrics
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The pith

Scattering maps and Poisson operators for two Schrödinger operators differ by a compact operator at a critical level if and only if their potentials are identical.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a uniqueness result for an inverse problem: on a fixed curved space belonging to a suitable class, the scattering map of the time-dependent Schrödinger equation recovers the potential uniquely. The scattering map sends the asymptotic profile of a solution at past infinity to its profile at future infinity. The key criterion is that two such maps, together with their associated Poisson operators, differ by a compact operator precisely when the potentials coincide. A sympathetic reader would care because this supplies a concrete criterion, expressed in operator-theoretic terms, that distinguishes one potential from another without requiring direct access to the interior of the space.

Core claim

For a certain class of metrics, the scattering maps and Poisson operators associated to two Schrödinger operators on the same curved space only differ by a compact operator on a critical level if and only if the two potentials are equal.

What carries the argument

The scattering map, which sends the asymptotic profile of solutions as time tends to negative infinity to the profile as time tends to positive infinity, together with the Poisson operators; the criterion that these objects differ by a compact operator at a critical level.

If this is right

  • The potential is uniquely determined by the scattering map under the given compactness condition.
  • Any two Schrödinger operators whose scattering data agree up to compact perturbation at the critical level must share the same potential.
  • The result supplies a rigidity statement for the time-dependent inverse scattering problem on curved spaces.
  • The equivalence extends directly to the associated Poisson operators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same compactness criterion might serve as a test for equality of potentials in numerical reconstructions of the scattering map.
  • If the class of metrics can be enlarged, the uniqueness statement could apply to a wider family of asymptotically flat or asymptotically hyperbolic spaces.
  • The result suggests examining whether analogous compactness conditions recover other coefficients, such as magnetic potentials or nonlinear terms, in related evolution equations.

Load-bearing premise

The underlying metrics belong to a class in which the scattering maps, Poisson operators, and the notion of differing by a compact operator at a critical level are all well-defined with the needed functional-analytic properties.

What would settle it

Exhibit two distinct potentials on a metric from the allowed class such that the associated scattering maps and Poisson operators differ by a compact operator at the critical level, or prove that no such pair exists.

read the original abstract

For a time dependent Schr\"odinger equation, the scattering map is the map sending the asymptotic profile of a solution as $t \to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for a certain class of metrics, the scattering maps and Poisson operators associated to two Schr\"odinger operators on the same curved space only differ by a compact operator on a critical level if and only if the two potentials are equal.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper proves an if-and-only-if uniqueness result for the time-dependent Schrödinger equation on curved spaces: for a certain class of metrics, the scattering maps and Poisson operators associated to two Schrödinger operators differ by a compact operator on a critical level if and only if the potentials are identical.

Significance. If the result holds, it provides a functional-analytic characterization of potentials from scattering data in the time-dependent setting, extending standard inverse-scattering uniqueness theorems to asymptotically controlled manifolds. The compact-difference condition at a critical level is a natural device for handling the relevant operators.

minor comments (1)
  1. The abstract refers to 'a certain class of metrics' without further specification; the introduction should explicitly delineate the functional-analytic assumptions (e.g., decay rates, curvature bounds) that guarantee the scattering map and Poisson operators are well-defined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the summary of our manuscript on determining potentials from the scattering map of the time-dependent Schrödinger equation. The recommendation is uncertain, but the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No circularity: direct uniqueness theorem on scattering maps

full rationale

The paper states a standard if-and-only-if uniqueness result for potentials recovered from scattering maps and Poisson operators differing by a compact operator at a critical level, restricted to a class of metrics where the objects are well-defined. No fitted parameters, self-definitional quantities, or load-bearing self-citations appear in the stated claim; the derivation chain is a mathematical proof of equivalence between two operators' data and the potentials, which is self-contained against external benchmarks and does not reduce any prediction to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on domain assumptions from scattering theory for Schrödinger operators on Riemannian manifolds; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Existence and functional properties of scattering maps and Poisson operators for the time-dependent Schrödinger equation on a certain class of metrics
    The result is stated to hold only for a certain class of metrics, implying reliance on these background properties being well-defined.

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Reference graph

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