Pith. sign in

REVIEW 2 major objections 5 minor 130 references

Inverse scattering problems for non-linear wave equations on Lorentzian manifolds

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that nonlinear scattering functionals uniquely determine the topology, differentiable structure, and conformal type of a globally hyperbolic Lorentzian manifold with an asymptotically Minkowskian infinity, and recover the…

desk verdict Genuinely new inverse scattering result with a repairable but load-bearing gap in Theorem 3 (Eq. (99)). read the letter →

arxiv 2411.09354 v2 pith:GTWIL22F submitted 2024-11-14 math.AP

Spyros Alexakis , Hiroshi Isozaki , Matti Lassas , Teemu Tyni This is my paper · ORCID
classification math.AP MSC 35R3035L0535L7153C50
keywords inversescatteringsemilinearwaveequationLorentzianmanifoldsfunctionalsconformaltypenonlinearinteractionofwavesPenrosecompactificationasymptoticallyMinkowskianinfinity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves an inverse scattering uniqueness theorem for the semilinear wave equation $\Box_g u + d u + a u^\kappa = 0$ with $\kappa \geq 4$ on a globally hyperbolic Lorentzian manifold that has at least one asymptotically Minkowskian infinity visible from the whole spacetime. The data are nonlinear scattering functionals: for each small incoming radiation field supported on a compact piece of past null infinity and each observation point $q$ on future null infinity, one records the value of the outgoing radiation field at $q$. The paper shows that these functionals determine the topology, differentiable structure, and conformal type of the spacetime, and that the metric and the nonlinear coefficient $a$ are recovered up to the rescaling $g \mapsto e^{2\gamma}g$, $a \mapsto e^{(\kappa-3)\gamma}a$. This matters because the scattering functionals remain well defined even when incoming waves blow up before a classical scattering operator could be defined, and the corresponding inverse problem for the linear wave equation remains open.

What carries the argument

The load-bearing object is the scattering functional $S_{t_1,q}(h^-)=h^+(q)$, defined for incoming radiation fields $h^-$ in the class of concentrated waves as the value at a point $q$ of the future radiation field of the nonlinear Goursat problem with data on past null infinity. The argument's engine is the reduction of these far-field point evaluations to near-field source-to-solution maps on the extended Penrose spacetime $N_{\mathrm{ext}} = N \cup N^+ \cup N^-$; a central step is the analysis of $\kappa$-th-order interactions of four conormal waves, whose interaction produces a wave whose leading singularity is supported on the future light cone of the interaction point and whose principal symbol contains $A(q)$ as a factor. That structure lets the proof reconstruct earliest light observation sets and then the conformal type of the metric.

What would settle it

In Minkowski space with a small, positive, compactly supported coefficient $a$, take an incoming radiation field that is the past trace of a linear wave generated by a compact source in the non-physical past, and compare, to first order in the amplitude, the future trace of the nonlinear Goursat solution with the linear future trace. Their difference is essentially $-\int G(q,y) A(y) u_{\mathrm{lin}}(y)^\kappa\, dy$; if a suitable source makes this integral nonzero, the asserted equality $S_{t_1,q}(u|_{I^-})=u|_{I^+}$ fails and the reduction in Theorem 3 breaks.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: two Lorentzian manifolds with asymptotically Minkowskian infinities whose scattering functionals agree for all small incoming radiation fields are related by a diffeomorphism $\Psi$ and a conformal factor $e^{2\gamma}$, and their nonlinear coefficients satisfy $a^{(1)} = e^{(\kappa-3)\gamma}\Psi^* a^{(2)}$. In other words, far-field nonlinear scattering data uniquely determine the manifold's topology, its differentiable structure, the conformal class of its metric, and the nonlinear coefficient up to the gauge freedom in (34). The proof conformally compactifies the physical spacetime, glues non-physical past and future regions onto the compactification, and first shows that the scattering functionals determine source-to-solution maps on this extended spacetime; nonlinear interactions of waves inside the physical region then act as structured probes whose scattered light cones reveal the conformal geometry.

Load-bearing premise

The reduction to near-field measurements assumes that a scattering functional, which is defined through the nonlinear equation, acts like the identity on the incoming trace of a linear wave: it returns the linear future trace unchanged, even though the nonlinear term is present along the wave's path.

Editorial extensions

If this is right

  • Equal scattering functionals force a conformal diffeomorphism between the two spacetimes, so the causal and null-cone structure of the manifold is fully encoded in the far-field data.
  • Once a conformal gauge is chosen, the coefficient $a$ of the nonlinear term is uniquely determined, because the remaining freedom is exactly the multiplication by $e^{(\kappa-3)\gamma}$ accompanying the metric rescaling.
  • Scattering functionals provide meaningful inverse-scattering data in regimes where the classical scattering operator is not defined, including cases where some incoming waves lead to finite-time blow-up.
  • The reduction to near-field source-to-solution maps on the extended spacetime means that local-measurement reconstruction techniques can be applied to purely far-field scattering measurements.
  • For conformally FLRW spacetimes satisfying the no-particle-horizon condition, restricted scattering data already determine the metric and nonlinear coefficient in the reconstruction domain, a setting the paper connects to cosmological models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: because the theorem recovers only the conformal class, any practical reconstruction must fix a gauge, for example by normalizing the metric at one point or prescribing the conformal factor; the theorem itself does not address that choice.
  • Beyond the paper: the scattering functionals are point evaluations of the outgoing field, so the result suggests that a dense family of point receivers at future null infinity carries the same information as the full radiation field, which would ease the demands on scattering experiments.
  • Beyond the paper: the closing sketch indicates the same strategy may extend to positive-ADM-mass, Schwarzschild-like spacetimes and cubic nonlinearities; if carried out, inverse scattering would reach more physical black-hole exteriors, where the direct problem is already harder.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces a new type of scattering data, called non-linear scattering functionals, for semilinear wave equations on globally hyperbolic Lorentzian manifolds with an asymptotically Minkowskian infinity. It claims (Theorem 2) that these functionals uniquely determine the topology, differentiable structure, and conformal type of the manifold, as well as the Lorentzian metric and the coefficient of the nonlinearity up to the multiplicative transformations in (34). The proof strategy is to reduce the scattering problem to a near-field problem on an extended manifold obtained by gluing non-physical regions N^+ and N^- to the Penrose compactification, and then to apply the higher-order linearization method from previous work [75, 83]. Theorem 3 is the bridge that converts equality of scattering functionals into equality of source-to-solution maps, and the rest of the paper develops the microlocal analysis of nonlinear interactions to reconstruct the conformal geometry and the coefficient a(x). The paper also contains examples (wormholes, locally Schwarzschild horizons, FLRW-type space-times) and a discussion of generalizations to non-smooth metrics, with details deferred to later work.

Significance. If the proof is repaired, this would be a substantial advance in inverse scattering for nonlinear hyperbolic equations on Lorentzian manifolds. The introduction of scattering functionals as data that remain well-defined even when classical scattering operators fail due to blow-up is a useful conceptual contribution, and the reduction from scattering data to near-field source-to-solution maps via Penrose compactification is an elegant strategy that extends the earlier local methods of [75, 83] to global inverse scattering. The paper also gives concrete and nontrivial examples of manifolds with several ends or event-horizon-like structures that fall within the scope of the theorem. The authors are explicit about the external benchmark results they rely on, and they acknowledge the complementary work of Hintz, Sá Barreto, Uhlmann, and Zhang [53].

major comments (2)
  1. [Section 3.2, Eqs. (93)-(100)] The equality S^{(j)}_{t1,q}(u^{(j)}|_{I^-})(q') = u^{(j)}|_{I^+}(t2)(q') is asserted without proof and is generally false. The scattering functional S_{t1,q} is defined in Definition 4 as the value at q of the solution of the nonlinear Goursat problem (21)-(23), which contains the term A e{u}^\kappa. The function u^{(j)} appearing in (93) solves the linear equation (□_{gext}+B)u^{(j)} = f, not the nonlinear equation. For an incoming trace h^- of small amplitude ε, the nonlinear solution with this past radiation field differs from the linear solution by a term of order ε^\kappa when A is nonzero. Thus the identification in (99) is not a consequence of the definitions; it would only hold in the trivial case A ≡ 0. Since this equality is the only step that transfers the assumed equality of scattering functionals to equality of the future traces of the waves used to construct the near-field source-to-solution maps, Theorem 3 is not proved as written.
  2. [Section 3.2, Eqs. (93)-(100)] A repair of the gap in Eq. (99) is available but requires a nontrivial modification of the proof. One should replace the linear problem (93) by the nonlinear problem (40), i.e., solve (□_{gext}+B)w + A w^\kappa = f with supp(w) ⊂ J^+(supp(f)). Because A and B vanish on N^-, the solution w in N^- is the same standard linear wave for both manifolds, so the traces w^{(1)}|_{I^-} and w^{(2)}|_{I^-} coincide and lie in B^-(R(t1)) with sufficiently small norm by Theorem 5(ii). The assumed equality of scattering functionals then gives equality of w^{(1)}|_{I^+} and w^{(2)}|_{I^+}, and the Goursat problem in N^+ (now with the same nonlinear equation) yields equality of the source-to-solution maps L_{g^j,B_j,A_j,p^+,K_n}. The submitted proof does not make this replacement, and the current line of argument is therefore not merely under-justified but incorrect at the displayed equality.
minor comments (5)
  1. [Section 1.5] In the sentence defining the compact sets K_n, the text reads 'compact sets that care closures of open sets'; the word 'care' should be 'are'.
  2. [Definition 4] The symbol ε is used both for the radius of the domain D^{(ε)}(S_{t1,q}) in (33) and as the smallness parameter in Theorem 5; the relation between the two (for instance, the choice of ε(t1,q1) in the sentence after (33)) would be clearer if denoted by different letters.
  3. [Section 2.0.2, proof of Lemma 1] In the proof that Σ∪{i0} is a Cauchy surface of Next, the assertion that J^-_{Next}(x1) ∩ J^+_{Next}(i0) is compact is used without proof; this is not immediate from the definition of Next and should be justified or replaced by an explicit argument.
  4. [Section 5.0.1] The subsection 'On the further generalizations' contains a sketch for metrics that are only C^0 near i0 and states that details will be presented elsewhere; this is acceptable as a remark, but it should be clearly labeled as work in progress rather than as part of the main theorem, since the main theorem is restricted to the smooth case.
  5. [Theorem 2] The statement of Theorem 2 and the transformation (34) do not mention the linear potential d(x) that appears in the equation (18). The authors might add a remark explaining whether d is also recovered, or whether it is an obstruction to uniqueness that the statement intentionally omits.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the scattering functionals are genuine data and the reduction to near-field maps, though containing a correctness gap at Eq. (99), does not assume the theorem's conclusion.

full rationale

Walking the claimed chain — Definition 4, Theorem 3, Lemma 6, and Steps 2–6 of Theorem 2 — I find no place where the output is presupposed in the input. The scattering functionals S_{t1,q}(h^-) are defined as point evaluations of the future radiation field of the solution of the nonlinear Goursat problem (21)–(23); the inverse problem then asks whether this data determines (M,g,a). No parameter is fitted to a subset of that data and then reported as a prediction, and the final recovery step is delegated to the published near-field theorems [75,83], whose assumptions (given source-to-solution map, known conformal observation sets) do not contain the scattering data or the conclusion of this paper. Although [75,83] share authors with the present paper, that is normal self-citation and not circular, because the cited results are independent published theorems rather than restatements of the present claim. The only load-bearing defect I located is in the proof of Theorem 3, Eq. (99): the paper asserts S^{(j)}_{t1,q}(u^{(j)}|_{I^-}) = u^{(j)}|_{I^+} for the solution u^{(j)} of the linear problem (93). This does not follow from Definition 4, which evaluates S on the nonlinear Goursat solution, and it is generally false when A is nonzero. That is a correctness gap in the submitted proof, repairable by solving the nonlinear equation (40) instead, but it is not circularity: it does not make the theorem's conclusion equivalent to its hypotheses. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim introduces no free parameters; the only new object is the scattering functional data, which are measurements, and the extended space-time Next is a mathematical device without independent physical evidence. The result depends on the completeness of the scattering data (all t1,q), on the non-vanishing of the nonlinear coefficient, and on several substantial prior results, most notably Kurylev-Lassas-Uhlmann [75] and Lassas-Uhlmann-Wang [83], which are used as black boxes. The paper also assumes smoothness of the conformal compactification at the spatial infinity i0, explicitly deferring the non-smooth (e.g., physical black hole) case to future work.

assumptions (8)
  • domain assumption The manifold (M,g) is globally hyperbolic with an asymptotically Minkowskian infinity E visible in the whole space-time (J^+(E) = J^-(E) = M).
    Assumed in Definition 2 and Theorem 2; needed for the scattering functionals to be defined and for the causality arguments.
  • domain assumption The coefficient a of the nonlinear term is nowhere vanishing on M, and a and d are Schwartz at infinity.
    a(x) != 0 is required in Lemma 6 and the reconstruction of light observation sets; if a vanished on a set, interactions there would produce no signal.
  • domain assumption kappa >= 4 is an integer.
    The proof uses kappa-th order interactions with kappa >= 4 to avoid 3-wave interaction singularities; the microlocal analysis assumes this.
  • domain assumption The scattering functionals are known for all -pi < t1 < 0 and q in I^+, i.e., the data are complete.
    Theorem 2 assumes equality of these functionals for all such parameters; the reconstruction of the full source-to-solution maps uses this infinite data set.
  • domain assumption The conformal compactification has a single spatial infinity i0 (Penrose-type), and Next is smooth near i0.
    The construction of the extended space-time Next and the global hyperbolicity Lemma 1 rely on smoothness at i0; non-smooth (e.g., Schwarzschild-like) cases are deferred to future work (Section 5.0.1).
  • standard math Existence and uniqueness for the Goursat-Cauchy problem for the nonlinear wave equation on Lipschitz domains in globally hyperbolic manifolds.
    Theorem 5 and Appendix A provide energy estimates; referenced results [94,101,29] are used.
  • standard math Strong Huygens' principle on R x S^3, and the Lagrangian intersection calculus for conormal distributions [83].
    Used to show that scattering functionals are well-defined and to compute principal symbols of interaction waves.
  • standard math The near-field inverse problem for Lorentzian manifolds from source-to-solution maps (Kurylev-Lassas-Uhlmann [75, Theorem 1.2]).
    The final step of Theorem 2 applies [75] to reconstruct the conformal type from light observation sets; not re-proven in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inverse scattering problems for non-linear wave equations on Lorentzian manifolds." pith.science (2026). https://pith.science/paper/GTWIL22F

@misc{pith2026241109354,
  author       = {Pith},
  title        = {Pith review of: Inverse scattering problems for non-linear wave equations on Lorentzian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTWIL22F}},
  note         = {Machine review of arXiv:2411.09354}
}
read the original abstract

We show that an inverse scattering problem for a semilinear wave equation can be solved on a manifold having an asymptotically Minkowskian infinity, that is, scattering functionals determine the topology, differentiable structure, and the conformal type of the manifold. Moreover, the metric and the coefficient of the non-linearity are determined up to a multiplicative transformation. The manifold on which the inverse problem is considered is allowed to be an open, globally hyperbolic manifold which may have non-trivial topology or several infinities (i.e., ends) of which at least one has to be of the asymptotically Minkowskian type. To formulate the inverse problems we define a new type of data, non-linear scattering functionals, which are defined also in the cases where the classically defined scattering operators are not well-defined. This makes it possible to solve inverse problems also in cases where some of the incoming waves lead to a blow-up of the scattered solution. We use non-linear interaction of waves as a beneficial tool that helps to solve the inverse problem. The corresponding inverse problem for the linear wave equation still remains unsolved.

Figures

Figures reproduced from arXiv: 2411.09354 by the authors.

Figure 1
Figure 1. Left: The Penrose map is a conformal map Φ : R×R 3 → R×S 3 and its image Nb = Φ(R×R 3 ) ⊂ R×S 3 is the Penrose compactification of the Minkowski space, see (6). In the figure R × S 3 is visualized as a cylindrical surface R × S 1 , and Nb is visualized as the area shaded by the red lines, that is, Nb is visualized as a subset that is cut from the cylinder by two “circles”, one of which passes through the points i0 a… view at source ↗
Figure 2
Figure 2. Visualization of the definition of the asymptot￾ically Minkowskian infinity E ⊂ M. The figures show the Penrose diagrams that are 2-dimensional analogs of the cylin￾ders shown in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Left: The set P(R) is shown as the horizontal bold gray line. The grayed region depicts the set S(R), while its restriction S−(R) to the past null infinity I − is shown as the diagonal gray line. Right: Sets and the support of the cut-off function ρ used in the proof of Theorem 5. be the image of the set {0}×BR3 (0, R) under the Penrose map Φ. Moreover, let S(R) = {γx,ξ(s) ∈ R × S 3 | x ∈ P(R), ξ ∈ Lx(R × S 3 ), s ∈… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Left: Visualization of setting where scatter￾ing functionals are defined. The in-going radiation field h− is supported on a relatively compact subset of I −. When ∥h−∥ < ε(t1, q1), the solution u of the scattering problem is defined in the past of the point q1. The sca…
Figure 5
Figure 5. Figure 5: Left: Morris-Thorne wormhole manifold, see [91,92], is a static universe (that is non-physical due to nega￾tive mass) of the form M = R×N0, where N0 is illustrated in the figure. Right: Penrose diagramm of a (non-physical) tra￾versable wormhole, see [11,43,125]. Note t…
Figure 6
Figure 6. Figure 6: We can consider product space-times M = R×N0 where N0 is a 3-dimensional Riemannian manifold with non￾trivial topology or several ends. One of the ends of N0 has to be asymptotically Euclidean so that the manifold M has an asymptotically Minkowskian infinity. The figur…
Figure 7
Figure 7. Figure 7: Left. The figure shows the (1 + 2) dimensional spacetime R×(R 2 \BR2 (0, rs)) that is analogous to the (1+3) dimensional spacetime M0 = R × (R 3 \ BR3 (0, rs)), see Ex￾ample 3. In this space-time, we consider the metric grs given in (36). The surface r = rs is visualiz…
Figure 8
Figure 8. Figure 8: Left: Penrose diagram. The red curves corre￾spond to 3-dimensional light-like surfaces I − and I +. Note that in the case when the metric g in R 3 × R is time￾independent, the metric ge may be non-smooth near the points i+ = (π, {NP}) and i− = (−π, {NP}). Also, i0 = (0…
Figure 9
Figure 9. Figure 9: The directed conormal sources εjfj , supported in N− near the past light-like infinity I −, produce distorted plane waves uj . The left and the center left figures show, at the times t1 and t2 the singular supports of such waves. When the four waves have non-linear int…
Figure 10
Figure 10. Figure 10: Subsets W = I +(p − 0 ) ∩ I −(p+2) and V = I −(p + 0 ) ∩ I +(p−2) of Next in the Penrose compactification. Moreover, we define the domain DV1,V2 = D(Lg,B,A;V1,V2 ) of the map Lg,A,B;V1,V2 to be the union D(g, A, B; V1, V2) = [ K⊂⊂V1 {f ∈ Hk 0 (K) | ∥f∥Hk(V1) < εK}, wh…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

130 extracted references · 80 canonical work pages

  1. [75]

    Kurylev, M

    Y. Kurylev, M. Lassas, and G. Uhlmann. Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations.Invent. Math., 212(3):781–857, 2018

    work page 2018

  2. [83]

    Lassas, G

    M. Lassas, G. Uhlmann, and Y. Wang. Inverse problems for semilinear wave equa- tions on lorentzian manifolds.Comm. Math. Phys., 360:555–609, 2018

    work page 2018

  3. [53]

    Hintz, A

    P. Hintz, A. S. Barreto, G. Uhlmann, and Y. Zhang. Inverse nonlinear scattering by a metric, 2024

    work page 2024

  4. [1]

    Acosta, G

    S. Acosta, G. Uhlmann, and J. Zhai. Nonlinear ultrasound imaging modeled by a westervelt equation.SIAM J. Appl. Math., 82(2022):558–607, 2022

    2022

  5. [2]

    Alexakis, A

    S. Alexakis, A. Feizmohammadi, and L. Oksanen. Lorentzian Calderón problem under curvature bounds.Invent. Math., 229(1):87–138, 2022

    2022

  6. [3]

    Alexakis, A

    S. Alexakis, A. Feizmohammadi, and L. Oksanen. Lorentzian calderon problem near the minkowski geometry.J. Eur. Math. Soc., 2024

    2024

  7. [4]

    P. Alinhac. Non-unicite du probleme de cauchy.Ann. of Math., 117:77–108, 1983

    1983

  8. [5]

    Aretakis

    S. Aretakis. General relativity, University of Toronto, 2018. Lecture notes

    2018

Show all 130 references
  1. [6]

    Y. M. Assylbekov and T. Zhou. Direct and inverse problems for the nonlinear time- harmonic maxwell equations in kerr-type media.J. Spectr. Theory, 11(1):1–38, 2021

    2021

  2. [7]

    Y. M. Assylbekov and T. Zhou. Inverse problems for nonlinear maxwell’s equations with second harmonic generation.J. Differential Equations, 296:148–169, 2021

    2021

  3. [8]

    J. C. Baez, I. E. Segal, and Z.-F. Zhou. The global goursat problem and scattering for nonlinear wave equations.Journal of Functional Analysis, 93(2):239–269, 1990

    1990

  4. [9]

    J. C. Baez and Z.-F. Zhou. Analyticity of scattering for theϕ4 theory.Comm. Math. Phys., 124(1):9–21, 1989

    1989

  5. [10]

    Balehowsky, A

    T. Balehowsky, A. Kujanpää, M. Lassas, and T. Liimatainen. An inverse problem for the relativistic Boltzmann equation.Comm. Math. Phys., 396(3):983–1049, 2022

    2022

  6. [11]

    Bambi and D

    C. Bambi and D. Stojkovic. Astrophysical wormholes.Universe, 7:136, 2021

    2021

  7. [12]

    C. Bär, N. Ginoux, and F. Pfäffle. Wave equations on lorentzian manifolds and quantization. ESI Lectures in Mathematics and Physics, 202 pp, 2007

    2007

  8. [13]

    J. A. Barceló, D. Faraco, A. Ruiz, and A. Vargas. Reconstruction of singularities from full scattering data by new estimates of bilinear fourier multipliers.Math. Ann., 346(3):505–544, 2010

    2010

  9. [14]

    Baskin and A

    D. Baskin and A. Sá Barreto. Radiation fields for semilinear wave equations.Trans. Amer. Math. Soc., 367(6):3873–3900, 2015

    2015

  10. [15]

    Baskin, A

    D. Baskin, A. Vasy, and J. Wunsch. Asymptotics of scalar waves on long-range asymptotically Minkowski spaces.Adv. Math., 328:160–216, 2018

    2018

  11. [16]

    Baskin and F

    D. Baskin and F. Wang. Radiation fields on Schwarzschild spacetime.Comm. Math. Phys., 331(2):477–506, 2014

    2014

  12. [17]

    Berezanskii

    Y. Berezanskii. The uniqueness theorem in the inverse problem of spectral analysis for the schrödinger equation.Trudy Moskov Mat. Obs., 7:1–62, 1958

    1958

  13. [18]

    Bernal and M

    A. Bernal and M. Sanchez. Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes.Comm. Math. Phys., 257:43–50, 2005

    2005

  14. [19]

    causal” instead of “strongly causal

    A. Bernal and M. Sanchez. Globally hyperbolic spacetimes can be defined as “causal” instead of “strongly causal".Class. Quant. Grav., 24:745–750, 2007

    2007

  15. [20]

    Beyer, J

    F. Beyer, J. Frauendiener, and J. Hennig. Explorations of the infinite regions of spacetime. Int. J. Mod. Phys. D, 29, 2020. Article 2030007

    2020

  16. [21]

    Cakoni and D

    F. Cakoni and D. Colton.Qualitative methods in inverse scattering theory. An in- troduction.Springer, viii+277 pp, 2006

    2006

  17. [22]

    A qualitative approach to inverse scattering theory.Applied Mathematical Sciences, 188

    F.CakoniandD.Colton. A qualitative approach to inverse scattering theory.Applied Mathematical Sciences, 188. Springer, New York, x+297 pp, 2014

    2014

  18. [23]

    Cakoni, D

    F. Cakoni, D. Colton, and H. Haddar. Inverse scattering theory and transmission eigenvalues. pages x+193 pp. SIAM, 2016

    2016

  19. [24]

    C. I. Carstea, A. Feizmohammadi, Y. Kian, K. Krupchyk, and G. Uhlmann. The calderón inverse problem for isotropic quasilinear conductivities.Adv. Math., 391,

  20. [25]

    Cattoën and M

    C. Cattoën and M. Visser. Cosmodynamics: energy conditions, Hubble bounds, den- sity bounds, time and distance bounds.Classical Quantum Gravity, 25(16):165013, 19, 2008

    2008

  21. [26]

    I. Chavel. Riemannian Geometry: A Modern Introduction. Cambridge University Press, Cambridge Studies in Advanced Mathematics, 2nd edition, 2006

    2006

  22. [27]

    X. Chen, M. Lassas, L. Oksanen, and G. Paternain. Inverse problem for the yang- mills equations.Commun. Math. Phys., 384(2):1187–1225, 2021. INVERSE SCATTERING IN LORENTZIAN MANIFOLDS 65

    2021

  23. [28]

    X. Chen, M. Lassas, L. Oksanen, and G. Paternain. Detection of Hermitian con- nections in wave equations with cubic non-linearity.J. Eur. Math. Soc. (JEMS), 24(7):2191–2232, 2022

    2022

  24. [29]

    Choquet-Bruhat

    Y. Choquet-Bruhat. General relativity and the Einstein equations. Oxford Univ. Press, xxvi+785 pp, 2009

    2009

  25. [30]

    Classical Quantum Gravity, 19(9):L71–L79, 2002

    P.T.ChruścielandE.Delay.Existenceofnon-trivial, vacuum, asymptoticallysimple spacetimes. Classical Quantum Gravity, 19(9):L71–L79, 2002

    2002

  26. [31]

    Corvino and R

    J. Corvino and R. M. Schoen. On the asymptotics for the vacuum Einstein constraint equations. J. Differential Geom., 73(2):185–217, 2006

    2006

  27. [32]

    Daudé, N

    T. Daudé, N. Kamran, and F. Nicoleau. Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets.Ann. Inst. Fourier (Greno- ble), 69(1):119–170, 2019

    2019

  28. [33]

    de Hoop, G

    M. de Hoop, G. Uhlmann, and Y. Wang. Nonlinear interaction of waves in elasto- dynamics and an inverse problem.Math. Ann., 376(1-2):765–795, 2020

    2020

  29. [34]

    M. V. de Hoop, G. Uhlmann, and A. Vasy. Recovery of material parameters in transversely isotropic media.Arch. Ration. Mech. Anal., 235(1):141–165, 2020

    2020

  30. [35]

    Duistermaat.Fourier Integral Operators

    J. Duistermaat.Fourier Integral Operators. Birkhäuser, 1996

    1996

  31. [36]

    Eptaminitakis and P

    N. Eptaminitakis and P. Stefanov. Weakly nonlinear geometric optics for the Wester- velt equation and recovery of the nonlinearity.SIAM J. Math. Anal., 56(1):801–819, 2024

    2024

  32. [37]

    G. Eskin. Inverse problems for general second order hyperbolic equations with time- dependent coefficients.Bull. Math. Sci., 7(2):247–307, 2017

    2017

  33. [38]

    Feizmohammadi, Y

    A. Feizmohammadi, Y. Kian, and G. Uhlmann. An inverse problem for a quasilinear convection–diffusion equation.Nonl. Anal., 222, 2022

    2022

  34. [39]

    Feizmohammadi, M

    A. Feizmohammadi, M. Lassas, and L. Oksanen. Inverse problems for non-linear hyperbolic equations with disjoint sources and receivers.Forum math. Pi, 10:52, 2021

    2021

  35. [40]

    Feizmohammadi and L

    A. Feizmohammadi and L. Oksanen. An inverse problem for a semi-linear elliptic equation in riemannian geometries.J. Diff. Eq., 269(6):4683–4719, 2020

    2020

  36. [41]

    D. D. S. Ferreira, C. Kenig, M. Salo, and G. Uhlmann. Limiting Carleman weights and anisotropic inverse problems.Inventiones mathematicae, 178(1):119–171, 2009

    2009

  37. [42]

    Galarce, D

    F. Galarce, D. Lombardi, and O. Mula. Reconstructing haemodynamics quantities of interest from Doppler ultrasound imaging.Int. J. Numer. Methods Biomed. Eng., 37(2):Paper No. e3416, 34, 2021

    2021

  38. [43]

    Garattini and F

    R. Garattini and F. Lobo. Self-sustained traversable wormholes in noncommutative geometry.Physics Letters B, 671:146–152, 2009

    2009

  39. [44]

    R. Geroch. Domain of dependence.J. Math. Phys., 11:437–449, 1970

    1970

  40. [45]

    Greenleaf and G

    A. Greenleaf and G. Uhlmann. Recovering singularities of a potential from singular- ities of scattering data.Comm. Math. Phys., 157(3):549–572, 1993

    1993

  41. [46]

    Guillarmou, M

    C. Guillarmou, M. Salo, and L. Tzou. Inverse scattering at fixed energy on surfaces with Euclidean ends.Comm. Math. Phys., 303(3):761–784, 2011

    2011

  42. [47]

    Guillarmou and L

    C. Guillarmou and L. Tzou. Calderón inverse problem with partial data on Riemann surfaces. Duke Mathematical Journal, 158(1):83–120, 2011

    2011

  43. [48]

    Guillemin and G

    V. Guillemin and G. Uhlmann. Oscillatory integrals with singular symbols.Duke Math. J., 48:251–267, 1981

    1981

  44. [49]

    Haláček and T

    J. Haláček and T. Ledvinka. Analytical conformal compactification of schwarzschild spacetime. Relativity and Gravitation. Springer, 157, 2014

    2014

  45. [50]

    Hassell, R

    A. Hassell, R. Melrose, and A. Vasy. Spectral and scattering theory for symbolic potentials of order zero.Adv. Math., 181(1):1–87, 2004

    2004

  46. [51]

    Hassell, R

    A. Hassell, R. Melrose, and A. Vasy. Microlocal propagation near radial points and scattering for symbolic potentials of order zero.Anal. PDE, 1(2):127–196, 2008

    2008

  47. [52]

    P. Hintz. Black hole gluing in de Sitter space. Commun. Partial Differ. Equ., 46(7):1280–1318, 2021

    2021

  48. [54]

    Hintz and A

    P. Hintz and A. Vasy. Asymptotics for the wave equation on differential forms on Kerr-de Sitter space.J. Differential Geom., 110(2):221–279, 2018. 66 ALEXAKIS, ISOZAKI, LASSAS, TYNI

    2018

  49. [55]

    Hora and A

    R. Hora and A. Sá Barreto. Inverse scattering with partial data on asymptotically hyperbolic manifolds.Anal. PDE, 8(3):513–559, 2015

    2015

  50. [56]

    Hora and A

    R. Hora and A. Sá Barreto. Inverse scattering with disjoint source and observation sets on asymptotically hyperbolic manifolds.Comm. Partial Differential Equations, 43(9):1363–1376, 2018

    2018

  51. [57]

    Hörmander.The analysis of linear partial differential operators I

    L. Hörmander.The analysis of linear partial differential operators I. Springer-Verlag, viii+525 pp, 1985

    1985

  52. [58]

    Hörmander

    L. Hörmander. The analysis of linear partial differential operators IV. Springer- Verlag, vii+352 pp, 1985

    1985

  53. [59]

    Hörmander

    L. Hörmander. A remark on the characteristic cauchy problem.J. Funct. Anal., 93:270–277, 1990

    1990

  54. [60]

    Hughes, T

    T. Hughes, T. Kato, and J. Marsden. Well-posed quasi-linear second-order hyper- bolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal., 63:273–294, 1976

    1976

  55. [61]

    Isozaki.Many-body Schrödinger equation–scattering theory and eigenfunction ex- pansions

    H. Isozaki.Many-body Schrödinger equation–scattering theory and eigenfunction ex- pansions. Mathematical Physics Studies. Springer, Singapore, 2023

    2023

  56. [62]

    M. S. Joshi and A. Sá Barreto. Recovering asymptotics of metrics from fixed energy scattering data.Invent. Math., 137(1):127–143, 1999

    1999

  57. [63]

    M. S. Joshi and A. Sá Barreto. Inverse scattering on asymptotically hyperbolic manifolds. Acta Math., 184(1):41–86, 2000

    2000

  58. [64]

    Joudioux

    J. Joudioux. Conformal scattering for a nonlinear wave equation.J. Hyperbolic Dif- fer. Equ., 9(1):1–65, 2012

    2012

  59. [65]

    Joudioux

    J. Joudioux. Hörmander’s method for the characteristic Cauchy problem and con- formal scattering for a nonlinear wave equation.Lett. Math. Phys., 110:1391, 2020

    2020

  60. [66]

    T. Kato. Quasi-linear equations of evolution, with applications to partial differential equations. In L. N. in, editor,Spectral theory and differential equations, pages 25–70. Math, Vol. 448, Springer, 1975

    1975

  61. [67]

    Kenig and M

    C. Kenig and M. Salo. The Calderón problem with partial data on manifolds and applications. Analysis & PDE, 6(8):2003–2048, 2014

    2003

  62. [68]

    Kenig and M

    C. Kenig and M. Salo. Recent progress in the Calderón problem with partial data. Contemp. Math, 615:193–222, 2014

    2014

  63. [69]

    W. H. Kinney. Cosmology, inflation and the physics of nothing. In H. B. Prosper andM.Danilov, editors, Techniques and Concepts of High-Energy Physics XII,pages 189–243, Dordrecht, 2003. Springer Netherlands

    2003

  64. [70]

    Krupchyk and G

    K. Krupchyk and G. Uhlmann. Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities.Math. Res. Lett., 27(6):1801–1824, 2020

    2020

  65. [71]

    Krupchyk and G

    K. Krupchyk and G. Uhlmann. A remark on partial data inverse problems for semi- linear elliptic equations.Proc. Amer. Math. Soc., 148(2):681–685, 2020

    2020

  66. [72]

    Krupchyk and G

    K. Krupchyk and G. Uhlmann. Inverse problems for nonlinear magnetic Schrödinger equations on conformally transversally anisotropic manifolds. Anal. PDE, 16(8):1825–1868, 2023

    2023

  67. [73]

    Krupchyk, G

    K. Krupchyk, G. Uhlmann, and L. Yan. A remark on inverse problems for nonlinear magnetic Schrödinger equations on complex manifolds. Proc. Amer. Math. Soc., 152(6):2413–2422, 2024

    2024

  68. [74]

    Kurylev, M

    Y. Kurylev, M. Lassas, L. Oksanen, and G. Uhlmann. Inverse problem for Einstein- scalar field equations.Duke Math. J., 171(16):3215–3282, 2022

    2022

  69. [76]

    R.-Y. Lai, G. Uhlmann, and Y. Yang. Reconstruction of the collision kernel in the nonlinear boltzmann equation.SIAM J. Math. Anal., 53(1):1049–1069, 2021

    2021

  70. [77]

    R.-Y. Lai, G. Uhlmann, and Y. Yang. Reconstruction of the collision kernel in the nonlinear Boltzmann equation.SIAM J. Math. Anal., 53(1):1049–1069, 2021

    2021

  71. [78]

    Lai and T

    R.-Y. Lai and T. Zhou. Partial data inverse problems for nonlinear magnetic Schrödinger equations.Math. Res. Lett., 30(5):1535–1563, 2023

    2023

  72. [79]

    M. Lassas. Inverse problems for linear and non-linear hyperbolic equations. InPro- ceedings of International Congress of Mathematicians ICM 2018, Vol III, pages 3739–3760, Rio de Janeiro, Brazil., 2018. INVERSE SCATTERING IN LORENTZIAN MANIFOLDS 67

    2018

  73. [80]

    Lassas, T

    M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo. Inverse problems for elliptic equations with power type nonlinearities.J. Math. Pures Appl., 145:44, 2021

    2021

  74. [81]

    Lassas, T

    M. Lassas, T. Liimatainen, L. Potenciano-Machado, and T. Tyni. Uniqueness, re- construction and stability for an inverse problem of a semi-linear wave equation.J. Diff. Eq., 337:395–435, 2022

    2022

  75. [82]

    Lassas, T

    M. Lassas, T. Liimatainen, L. Potenciano-Machado, and T. Tyni. Stability and lorentzian geometry for an inverse problem of a semi-linear wave equation.Anal. PDE., 2024. (in print)

    2024

  76. [84]

    Lax.Hyperbolic Partial Differential Equations

    P. Lax.Hyperbolic Partial Differential Equations. Courant lecture notes, AMS, 2006

    2006

  77. [85]

    Lax and R

    P. Lax and R. Phillips.Scattering theory, volume 26 ofPure and Applied Mathemat- ics. Academic Press, second edition, 1989

    1989

  78. [86]

    Liimatainen and L

    T. Liimatainen and L. Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Probl. Imaging, 16(2):467–479, 2022

    2022

  79. [87]

    R. Melrose. Geometric scattering theory. Stanford Lectures. Cambridge University Press, Cambridge, 1995

    1995

  80. [88]

    Melrose and G

    R. Melrose and G. Uhlmann. Lagrangian intersection and the cauchy problem. Comm. Pure Appl. Math., 32:483–519, 1979

    1979

  81. [89]

    Melrose and M

    R. Melrose and M. Zworski. Scattering metrics and geodesic flow at infinity.Invent. Math., 124(1-3):389–436, 1996

    1996

  82. [90]

    Geometric scattering theory.StanfordLectures.CambridgeUniversity Press, Cambridge, xii+116 pp, 1995

    R.B.Melrose. Geometric scattering theory.StanfordLectures.CambridgeUniversity Press, Cambridge, xii+116 pp, 1995

    1995

  83. [91]

    M. S. Morris and K. S. Thorne. Wormholes in spacetime and their use for interstellar travel. Am. J. Phys., 56:395, 1988

    1988

  84. [92]

    T. Muller. Exact geometric optics in a morris-thorne wormhole spacetime.Physical review D, 20:044043, 2008

    2008

  85. [93]

    Nakamura, M

    G. Nakamura, M. Vashisth, and M. Watanabe. Inverse initial boundary value prob- lem for a non-linear hyperbolic partial differential equation.Inverse Problems, 37(1),

  86. [94]

    J.-P. Nicolas. On Lars Hörmanders remark on the characteristic cauchy problem. Ann. de l’Institut Fourier, Tome, 56(3):517–543, 2006

    2006

  87. [95]

    015012, 27 pp

    Paper No. 015012, 27 pp

  88. [96]

    R. G. Novikov. The inverse scattering problem at fixed energy for the three- dimensional Schrödinger equation with an exponentially decreasing potential. Comm. Math. Phys., 161(3):569–595, 1994

    1994

  89. [97]

    R. G. Novikov. A multidimensional inverse spectral problem for the equation−∆ψ + (v(x) − Eu(x))ψ = 0. Funktsional. Anal. i Prilozhen., 22(4):11–22, 96, 1988

    1988

  90. [98]

    B. O’Neill. Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc., xiii+468 pp, 1983

    1983

  91. [99]

    Oksanen, M

    L. Oksanen, M. Salo, P. Stefanov, and G. Uhlmann. Inverse problems for real prin- cipal type operators.Amer. J. Math., 146(1):161–240, 2024

    2024

  92. [100]

    Petkov.Scattering Theory for Hyperbolic Operators

    V. Petkov.Scattering Theory for Hyperbolic Operators. North Holland, 1989

    1989

  93. [101]

    R. Penrose. Republication of: Conformal treatment of infinity.Gen. Relativity Grav- itation, 43(3):901–922, 2011

    2011

  94. [102]

    Sá Barreto, G

    A. Sá Barreto, G. Uhlmann, and Y. Wang. Inverse scattering for critical semilinear wave equations.Pure Appl. Anal., 4(2):191–223, 2022

    2022

  95. [103]

    Ringström.The Cauchy Problem in General Relativity

    H. Ringström.The Cauchy Problem in General Relativity. EMS, 2009

    2009

  96. [104]

    Sá Barreto and Y

    A. Sá Barreto and Y. Wang. Singularities generated by the triple interaction of semilinear conormal waves.Anal. PDE, 14(2021):1315–1367, 2020

    2021

  97. [105]

    Sá Barreto and Y

    A. Sá Barreto and Y. Wang. The scattering operator on asymptotically hyperbolic manifolds. J. Spectr. Theory, 9(1):269–313, 2019

    2019

  98. [106]

    Salo and X

    M. Salo and X. Zhong. An inverse problem for the p-Laplacian: boundary determi- nation. SIAM journal on mathematical analysis, 44(4):2474–2495, 2012

    2012

  99. [107]

    General relativity for mathematicians.GraduateTextsinMath- ematics, Vol

    R.SachsandH.Wu. General relativity for mathematicians.GraduateTextsinMath- ematics, Vol. 48. Springer-Verlag, New York-Heidelberg, 1977

    1977

  100. [108]

    Stefanov

    P. Stefanov. Inverse scattering problem for the wave equation with time-dependent potential. J. Math. Anal. Appl., 140(2):351–362, 1989

    1989

  101. [109]

    B. G. Schmidt. A new definition of conformal and projective infinity of space-times. Comm. Math. Phys., 36:73–90, 1974. 68 ALEXAKIS, ISOZAKI, LASSAS, TYNI

    1974

  102. [110]

    Stefanov, G

    P. Stefanov, G. Uhlmann, and A. Vasy. Boundary rigidity with partial data.J. Amer. Math. Soc., 29(2):299–332, 2016

    2016

  103. [111]

    Stefanov

    P. Stefanov. Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials.Math. Z., 201:541–559, 1989

    1989

  104. [112]

    Sun and G

    Z. Sun and G. Uhlmann. Inverse problems in quasilinear anisotropic media.Ameri- can Journal of Mathematics, 119(4):771–797, 1997

    1997

  105. [113]

    Stefanov, G

    P. Stefanov, G. Uhlmann, and A. Vasy. The transmission problem in linear isotropic elasticity.Pure Appl. Anal., 3(1):109–161, 2021

    2021

  106. [114]

    D. Tataru. Unique continuation for solutions to PDE’s; between Hörmander’s theo- rem and Holmgren’s theorem.Comm. PDE, 20:855–884, 1995

    1995

  107. [115]

    Sylvester and G

    J. Sylvester and G. Uhlmann. A global uniqueness theorem for an inverse boundary value problem.Annals of mathematics, pages 153–169, 1987

    1987

  108. [116]

    G. Uhlmann. Electrical impedance tomography and Calderón’s problem. Inverse problems, 25(12):123011, 2009

    2009

  109. [117]

    Taujanskas

    G. Taujanskas. Conformal scattering of the Maxwell-scalar field system on de Sitter space. J. Hyperbolic Differ. Equ., 16(4):743–791, 2019

    2019

  110. [118]

    Uhlmann and J

    G. Uhlmann and J. Zhai. On an inverse boundary value problem for a nonlinear elastic wave equation.J. Math. Pures Appl., 153(2021):1049–1069, 2021

    2021

  111. [119]

    Uhlmann and J

    G. Uhlmann and J. Zhai. Inverse problems for nonlinear hyperbolic equations.Dis- crete Contin. Dyn. Syst., 41(1):455–469, 2021

    2021

  112. [120]

    A. Vasy. Diffraction by edges.Modern Phys. Lett. B, 22:2287–2328, 2008

    2008

  113. [121]

    Uhlmann and Y

    G. Uhlmann and Y. Zhang. An inverse boundary value problem arising in nonlinear acoustics. SIAM J. Math. Anal., 55(2):1364–1404, 2023

    2023

  114. [122]

    A. Vasy. Microlocal analysis of asymptotically hyperbolic and kerr-de sitter spaces (with an appendix by s. dyatlov).Invent. Math., 194:381–513, 2013

    2013

  115. [123]

    A. Vasy. Propagation of singularities for the wave equation on manifolds with cor- ners. Ann. of Math., (2) 168:749–812, 2008

    2008

  116. [124]

    C. Villani. Synthetic theory of Ricci curvature bounds.Jpn. J. Math., 11(2):219–263, 2016

    2016

  117. [125]

    A. Vasy. Resolvents, Poisson operators and scattering matrices on asymptotically hyperbolic and de Sitter spaces.J. Spectr. Theory, 4(4):643–673, 2014

    2014

  118. [126]

    Wald.General Relativity

    R. Wald.General Relativity. University of Chicago Press, Chicago, 1984

    1984

  119. [127]

    M. Visser. Lorentzian Wormholes: From Einstein to Hawking. American Institute of Physics, 1996

    1996

  120. [128]

    Wang and T

    Y. Wang and T. Zhou. Inverse problems for quadratic derivative nonlinear wave equations. Comm. Partial Differential Equations, 44(11):1140–1158, 2019

    2019

  121. [131]

    Willett and J

    D. Willett and J. S. W. Wong. On the discrete analogues of some generalizations of Gronwall’s inequality.Monatsh. Math., 69:362–367, 1965. Spyros Alexakis, Department of Mathematics, University of Toronto, Canada. Hiroshi Isozaki, College of Mathematics, University of Tsukuba,...

    1965

  122. [2021]

    107956, 31 pp

    Paper No. 107956, 31 pp

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.