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Inverse spectral problems with sparse data and applications to passive imaging on manifolds

T0 review · 0 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A potential on a closed Riemannian manifold is uniquely determined by sparse spectral data—some eigenvalues and the eigenfunction traces on an open set—with no orthogonality or norming constants assumed.

desk verdict Genuinely new inverse spectral setup with real applications, but Lemma A.1's Tataru step is unjustified and looks load-bearing; the paper needs major revision before I'd trust it. read the letter →

arxiv 2507.22723 v1 pith:GGBTASAS submitted 2025-07-30 math.AP math.SP

Ali Feizmohammadi , Katya Krupchyk This is my paper · ORCID
classification math.APmath.SP MSC 35R3058J5035P05
keywords inversespectralproblemsparsedataSchrödingeroperatorpassivemeasurementgeometriccontrolconditionuniformobservabilityuniquecontinuationRiemannianmanifold
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 asks what a listener can recover when only some of the natural frequencies of a vibrating manifold and the shapes of the corresponding modes on a small open patch are observed. It proves two uniqueness theorems: a potential $V$ in the stationary Schrödinger operator $-\Delta_g+V$ on a closed Riemannian manifold is determined by such sparse spectral data, without assuming the eigenfunctions are orthogonal or globally normalized. The first theorem allows a large portion of the low- and high-frequency spectrum to be unused, subject to a geometric control condition on the observation patch; the second allows countably infinitely many spectral pairs to be missing for both operators, provided the eigenfunctions are uniformly observable from every open subset of the patch. The authors then turn this into a spectral framework for passive imaging, showing that a single measurement of the heat, Schrödinger, or wave equation can recover an unknown potential together with the unknown initial data.

What carries the argument

The proofs reduce the spectral matching problem to the wave equation. The carrying objects are: (i) the geometric control condition (GCC), or the stronger uniform observability estimate $\|\phi\|_{L^2(M)}\le C\|\phi\|_{L^2(O')}$ for eigenfunctions, which turns a finite observation window into control of the whole eigenfunction; (ii) the antipodal set $A_{M,g}(p)$, whose inclusion in $O$ lets finite-speed propagation and a global unique continuation theorem for distributional wave solutions upgrade agreement on $O$ to agreement on all of $M$; and (iii) a Paley–Wiener type interpolation theorem for uniformly discrete sets of zero upper density, which produces compactly supported time-window test functions whose Fourier transform vanishes on the present eigenvalue differences—this is the step that lets a countable infinity of spectral pairs be skipped.

What would settle it

A direct falsifier would be two potentials $V_1,V_2$ on the flat torus $\mathbb{T}^2$, with $V_2-V_1$ supported in a ball disjoint from a connected open patch $O$ containing antipodal points, whose eigenpairs satisfy (1.1) for all $k\ge N$; the theorem predicts no such pair exists, so any explicit construction would settle the matter.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: if two real-valued potentials $V_1,V_2$ on a closed connected Riemannian manifold share a subsequence of eigenvalues, $\mu_k^{(1)}=\mu_{b_k}^{(2)}$, and the corresponding eigenfunctions agree on an open connected observation set $O$ that satisfies the geometric control condition and contains the antipodal set of some point $p\in O$, then $V_1=V_2$ everywhere on $M$. Theorem 1.16 strengthens this to the case where countably many spectral pairs are entirely absent from the data for both operators, provided the eigenfunctions are observable from every nonempty open subset of $O$, with the missing eigenvalues quantified by a $\Lambda$-sparsity condition. Neither theorem requires the eigenfunctions to be orthonormal or any knowledge of global norming constants.

Load-bearing premise

The load-bearing premise of the stronger sparse-data theorem is the uniform observability condition (UO): the eigenfunctions of both Schrödinger operators must be observable from every nonempty open subset of the observation region with one uniform constant, a property currently known only for special geometries such as flat tori and Anosov surfaces.

Editorial extensions

If this is right

  • For any closed manifold and observation set satisfying (H), a potential is recoverable from partial spectral data even when a large portion of the low- and high-frequency spectrum is unused, with no eigenfunction normalization.
  • A single observation of the heat solution on $(0,\varepsilon)\times O$, for arbitrarily small $\varepsilon$, recovers both the potential and the initial temperature, under generic simplicity and nonzero Fourier coefficient assumptions for the first operator.
  • The same passive-measurement mechanism recovers the potential and the initial state for the time-dependent Schrödinger equation from $(0,\infty)\times O$, and for the wave equation recovers the potential plus both initial position and velocity.
  • On flat tori and Anosov surfaces, countably infinite holes in the spectral data still leave the potential uniquely determined.
  • When the manifold itself is unknown, the wave reduction recovers the manifold up to isometry from heat data on $(0,\varepsilon)\times O$, assuming generic spectral simplicity for both manifolds.

Reading between the lines

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

  • The $\Lambda$-sparsity condition is formulated purely in terms of eigenvalue locations, so the framework should apply to other self-adjoint elliptic operators on closed manifolds whenever an observability estimate and the corresponding unique continuation are available.
  • The residue arguments in the applications suggest a numerical passive-imaging algorithm: Laplace-transform a single heat trace on $(0,\varepsilon)\times O$ to read off eigenvalues and eigenfunction traces, then feed those traces into a wave-based reconstruction; the paper does not develop this algorithm.
  • Theorems 1.20 and 1.29 impose generic conditions only on the first operator, an asymmetry that goes beyond the one-dimensional passive-measurement results; whether this asymmetry is necessary in higher dimensions is left open by the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. This paper studies inverse spectral problems for Schrödinger operators on closed Riemannian manifolds, with data consisting of a sparse set of eigenvalues and the restrictions of the corresponding eigenfunctions to an open observation set. The main results are Theorem 1.3, which proves potential recovery under the geometric control condition and an antipodal containment assumption (H), and Theorem 1.16, which allows infinitely many spectral pairs to be missing for both operators under a uniform observability assumption (UO) and a Beurling–Kahane sparsity condition. The proofs reduce the spectral problem to a wave-equation problem, use observability estimates and finite-speed propagation, and invoke a global unique continuation theorem; Theorem 1.16 additionally uses a Paley–Wiener interpolation theorem. The paper then applies these results to simultaneous recovery of coefficients and initial data from a single passive measurement for heat, Schrödinger, and wave equations, and to recovery of the manifold in the zero-potential case.

Significance. Assuming correctness, these are strong and interesting results. The non-normalized, non-orthogonal eigenfunction formulation and the sparsity notion are genuine novelties; prior results in dimensions at least two with partial spectral data and no norming constants appear scarce. The applications to passive imaging are a significant extension of the one-dimensional theory and are made possible by the new spectral theorems. The proofs are detailed and use standard tools carefully; the paper also provides concrete examples, such as flat tori and Anosov surfaces, where the strong uniform observability assumption holds. The reliance on the companion paper [30] is explicit and appropriate: the geometric lemma and the manifold-recovery theorem are cited, not rederived. I specifically checked the unique-continuation step in Lemma A.1 and around equation (3.55); the conclusion is consistent with finite-speed propagation because the observation set W is centered at p, so the target set lies in the influence region of W. The paper is likely to be influential in inverse spectral theory and passive inverse problems.

minor comments (4)
  1. [Appendix A, Lemma A.1; §3, (3.53)–(3.55)] The global unique continuation step is terse enough that a reader may worry it conflicts with finite speed of propagation; since W is defined as B_{varepsilon/2}(p) in (2.6), the target set tilde O = {x : dist_g(x,p) < dist_g(p,q) - varepsilon} is contained in the T-neighborhood of W, so the conclusion is consistent with the standard backward domain-of-dependence argument, and a one-sentence clarification would improve the presentation.
  2. [Section 2, proof of Theorem 1.3] The letter V denotes both a potential (V_1, V_2) and the geodesic ball V = B_{varepsilon/2}(q); this notational clash is confusing and should be removed by renaming the ball, for instance to mathcal V.
  3. [Throughout] The phrase 'global Tataru unique continuation' is cited as [65, Theorem 3.24], [96], [45]; it would be helpful to state the precise formulation used, namely that vanishing of a distributional solution on an open spacetime cylinder implies vanishing on the associated influence region, so the reader can verify the hypotheses.
  4. [Theorem 1.20, proof] In the Laplace transform step, the index sequence (l_k)_{k geq k_0} is asserted to be strictly increasing; this follows because mu^{(1)}_k are distinct for k geq k_0, but the sentence could be added explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the main derivations are conditional on external theorems and stated assumptions rather than reducing to their own inputs.

full rationale

The core inverse spectral theorems, Theorem 1.3 and Theorem 1.16, are proved by converting the partial spectral data on the observation set O into data for associated wave evolutions, and then applying external results: finite-speed propagation, Tataru-type unique continuation from the cited literature, observability estimates, and the Beurling-Kahane interpolation theorem. None of these steps fits a parameter to a subset of the target spectral data and then reports a closely related quantity as a prediction. The injectivity estimate in Proposition 3.3 uses the assumed uniform observability bound (UO) with constants that do not depend on the unknown potentials; the interpolation function h in the proof of Theorem 1.16 is constructed from the sparsity set Gamma, not from the potentials V1 and V2, and is therefore not a fitted input disguised as a result. The substantial use of the authors' previous work [30] is technical or black-box rather than circular: Lemma A.1 and Lemma A.2 reproduce arguments from [30, Theorem 1.11], and Theorem 1.23 invokes [30, Theorem 1.11] as an external theorem whose hypotheses are established independently from the heat measurement; that cited theorem is parameter-free and does not include the present paper's target conclusion. The skeptical concern about the application of Tataru's unique continuation in Lemma A.1 and around equation (3.55) is a potential correctness or validity issue about whether the cited global unique continuation theorem applies as stated; it is not an instance of the paper's output being equivalent to its input by construction. The sparsity condition (Definition 1.15), the geometric condition (H), and the uniform observability condition (UO) are genuine hypotheses rather than conclusions manufactured from the data. Accordingly, no circular step can be identified from the text, and the paper's central claims are self-contained relative to the external theorems it invokes.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no invented entities. The central claims rest on a collection of standard results from spectral theory (Weyl's law, Seeley's theorem), control theory (observability from GCC), unique continuation (elliptic UCP, Tataru's global UCP), interpolation (Beurling-Kahane), and the BC method (source-to-solution map, [87]). The paper's own hypotheses, (H) and (UO), are domain assumptions on the observation set that are clearly flagged and justified by examples; they are the most restrictive inputs. The genericity assumptions in the applications (simple eigenvalues, nonzero Fourier coefficients) are also stated explicitly.

assumptions (10)
  • standard math Geometric control condition (GCC) implies observability of Schrödinger eigenfunctions from O (Lemma 2.1, citing [67], [76], [9,10,85]).
    Used to control L2 norms of eigenfunctions on all of M by their L2 norms on O; this is the key quantitative input in Theorems 1.3 and 1.16.
  • standard math Beurling-Kahane interpolation theorem for uniformly discrete zero-density sets (Theorem 3.2, [16], [50]).
    Allows construction of compactly supported L2 functions whose Fourier transforms interpolate arbitrary l2 data on the set Gamma; essential for Proposition 3.3 and Lemma 3.5 in handling infinitely many missing eigenvalues.
  • standard math Tataru's global unique continuation theorem for wave equations ([65, Theorem 3.24], [96], [45]).
    Propagates vanishing of wave solutions from a time slab over a small ball to a geodesically determined region; used in Lemma A.1 and (3.55) to extend equalities from O to M.
  • standard math Unique continuation property for elliptic operators -Delta_g+V-lambda (eigenfunctions).
    Used in Lemmas 2.4-2.7 and elsewhere to infer global identities from local ones.
  • standard math Weyl's law for the eigenvalue counting function N(mu) ≤ C mu^{n/2} of -Delta_g+V.
    Gives the eigenvalue growth (2.17) used to prove convergence of spectral expansions and to construct Λ-sparse subsequences in Example 4.
  • standard math Finite speed of propagation and well-posedness for linear wave equations on closed manifolds ([97], [87]).
    Forms the bridge from spectral data to wave evolutions; used throughout Sections 3 and Appendix A.
  • domain assumption Assumption (H): O satisfies GCC and contains the antipodal set A_{M,g}(p) of some p∈O.
    A hypothesis of Theorem 1.3; without it the proof's use of Tataru's UCP to transfer data from O to M fails, and the 1D counterexamples in [43] show recovery is not generally possible.
  • domain assumption Assumption (UO): for both potentials V1,V2, eigenfunctions are observable from every nonempty open subset of O (Definition 1.11).
    A much stronger hypothesis of Theorem 1.16; currently verified for flat tori and Anosov surfaces, but not for generic manifolds.
  • standard math External theorem [87, Theorem 1.1]: equality of local source-to-solution maps for the wave equation determines the potential.
    Used in Proposition A.3 for the classical Gel'fand problem with orthonormalized data; this is the BC method for closed manifolds.
  • domain assumption Genericity results: simple eigenvalues for a residual set of potentials/metrics ([98], [2]) and residual Fourier coefficients (Remark 1.21).
    The applications to passive imaging (Theorems 1.20, 1.29, 1.30) require the first operator to have simple high eigenvalues and the initial data to have nonzero Fourier coefficients; these are generic, not universal.

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Cite this review

Pith. "Pith review of Inverse spectral problems with sparse data and applications to passive imaging on manifolds." pith.science (2026). https://pith.science/paper/GGBTASAS

@misc{pith2026250722723,
  author       = {Pith},
  title        = {Pith review of: Inverse spectral problems with sparse data and applications to passive imaging on manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGBTASAS}},
  note         = {Machine review of arXiv:2507.22723}
}
read the original abstract

Motivated by inverse problems with a single passive measurement, we introduce and analyze a new class of inverse spectral problems on closed Riemannian manifolds. Specifically, we establish two general uniqueness results for the recovery of a potential in the stationary Schr\"odinger operator from partial spectral data, which consists of a possibly sparse subset of its eigenvalues and the restrictions of the corresponding eigenfunctions to a nonempty open subset of the manifold. Crucially, the eigenfunctions are not assumed to be orthogonal, and no information about global norming constants is required. The partial data formulation of our inverse spectral problems is naturally suited to the analysis of inverse problems with passive measurements, where only limited observational access to the solution is available. Leveraging this structure, we establish generic uniqueness results for a broad class of evolutionary PDEs, in which both the coefficients and the initial or source data are to be recovered from knowledge of the solution restricted to a subset of spacetime. These results introduce a spectral framework for passive imaging and extend inverse spectral theory into a regime characterized by highly incomplete, yet physically realistic, data.

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Works this paper leans on

99 extracted references · 76 canonical work pages

  1. [30]

    Feizmohammadi, A., Krupchyk, K., Uhlmann, G., Calder´ on problem for the fractional Schr¨ odinger operator on closed Riemannian manifolds, arXiv preprint (2024)

    work page 2024

  2. [1]

    Encyclopaedia of Mathematical Sciences, 1994, vol 63

    Agranovich, M., Elliptic operators on closed manifolds , Partial Differential Equations VI. Encyclopaedia of Mathematical Sciences, 1994, vol 63. Springer, Berlin,

    1994

  3. [2]

    H., Genericity of simple eigenvalues for elliptic PDE’s , Proc

    Albert, J. H., Genericity of simple eigenvalues for elliptic PDE’s , Proc. Amer. Math. Soc. 48 (1975), 413–418

    1975

  4. [3]

    Alexakis, S., Feizmohammadi, A., Oksanen, L., Lorentzian Calder´ on problem under curvature bounds, Invent. math. 229 (2022), 87–138

    2022

  5. [4]

    Anantharaman, N., Maci` a, F., Semiclassical measures for the Schr¨ odinger equation on the torus, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 6, 1253–1288. 55

    2014

  6. [5]

    Anderson, M., Katsuda, A., Kurylev, Y., Lassas, M., Taylor, M., Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem , Invent. Math. 158 (2004) 261–321

    2004

  7. [6]

    Athreya, J., Dyatlov, S., Miller, N., Semiclassical measures for complex hyperbolic quotients , preprint 2024, https://arxiv.org/abs/2402.06477

    work page Pith review arXiv 2024

  8. [7]

    Avdonin, S., Mikhaylov, V., Ramdani, K., Reconstructing the potential for the one- dimensional Schr¨ odinger equation from boundary measurements, IMA Journal of Mathemat- ical Control and Information, 31(1):137-150, 2014

    2014

Show all 99 references
  1. [8]

    Bando, S., Urakawa, H., Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tohoku Math. J. 35 (1983), no. 2, 155–172

    1983

  2. [9]

    Bardos, C., Lebeau, G., Rauch, J., Controle et stabilisation dans les problemes hyperboliques , in Controlabilite Exacte Perturbations et Stabilisation de Systeme Distribues, Appendice II, Tome 1, Collection RMA, Masson, Paris, 1988

    1988

  3. [10]

    Control Optim

    Bardos, C., Lebeau, G., Rauch, J., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , SIAM J. Control Optim. 30 (1992), 1024–1065

    1992

  4. [11]

    Bellassoued, M., Choulli, M., Dos Santos Ferreira, D., Kian, Y., Stefanov, P., A Borg-Levinson theorem for magnetic Schr¨ odinger operators on a Riemannian manifold , Ann. Inst. Fourier (Grenoble) 71 (2021), no. 6, 2471–2517

    2021

  5. [12]

    AN SSSR 297 (1987) 524–527 (in Russian)

    Belishev, M., An approach to multidimensional inverse problems for the wave equation , Dokl. AN SSSR 297 (1987) 524–527 (in Russian)

    1987

  6. [13]

    Belishev, M., Recent progress in the boundary control method, Inverse Problems , 23 (2007), R1–R67

    2007

  7. [14]

    Partial Differential Equations, 17 (1992), 767–804

    Belishev, M., Kurylev Y., To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767–804

    1992

  8. [15]

    Belishev, M., Langemann, D., Mikhaylov, A., Mikhaylov, V., On an inverse problem in pho- toacoustics, Journal of Inverse and Ill-posed Problems, 27:241-254, 2019

    2019

  9. [16]

    volume II, 1989

    Beurling, A., Carleson, L., Interpolation for an interval in R1, The collected works of Arne Beurling. volume II, 1989

    1989

  10. [17]

    Bestimmung der Dif- ferentialgleichung durch die Eigenwerte , Acta Math

    Borg, G., Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Dif- ferentialgleichung durch die Eigenwerte , Acta Math. 78, (1946), 1–96

    1946

  11. [18]

    Bourgain, J., Burq, N., Zworski, M., Control for Schr¨ odinger operators on 2-tori: rough potentials, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1597–1628

    2013

  12. [19]

    PDE 15 (2022), no.2, 273–326

    Bosi, R., Kurylev, Y., Lassas, M., Reconstruction and stability in Gel’fand’s inverse interior spectral problem, Anal. PDE 15 (2022), no.2, 273–326

    2022

  13. [20]

    Bukhgeim, A., Klibanov, M., Global uniqueness of a class of multidimensional inverse prob- lems, Dokl. Akad. Nauk SSSR, 260(2):269-272, 1981

    1981

  14. [21]

    Burq, N., Zworski, M., Control for Schr¨ odinger operators on tori, Math. Res. Lett. 19 (2012), no. 2, 309–324

    2012

  15. [22]

    Partial Differential Equations 38 (2013), no

    Choulli, M., Stefanov, P., Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data, Comm. Partial Differential Equations 38 (2013), no. 3, 455–476

    2013

  16. [23]

    6, 1265–1273

    Croke, C., Sharafutdinov, V., Spectral rigidity of a compact negatively curved manifold, Topol- ogy 37 (1998), no. 6, 1265–1273

    1998

  17. [24]

    II , 455–485, Math

    Datchev, K., Hezari, H., Inverse problems in spectral geometry, Inverse problems and appli- cations: Inside Out. II , 455–485, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013

    2013

  18. [25]

    220 (2018), no

    Dyatlov, S., Jin, L., Semiclassical measures on hyperbolic surfaces have full support , Acta Math. 220 (2018), no. 2, 297–339. 56 FEIZMOHAMMADI AND KRUPCHYK

    2018

  19. [26]

    Dyatlov, S., Jin, L., Nonnenmacher, S., Control of eigenfunctions on surfaces of variable curvature, Journal of the American Mathematical Society 35 (2022), 361–465

    2022

  20. [27]

    Feizmohammadi, A., Fractional Calder´ on problem on a closed Riemannian manifold, Trans. Amer. Math. Soc. 377 (2024), no. 4, 2991–3013

    2024

  21. [28]

    Feizmohammadi, A., Reconstruction of 1-D evolution equations and their initial data from one passive measurement, SIAM Journal of Mathematical Analysis, in press 2025

    2025

  22. [29]

    Feizmohammadi, A., Kian, Y., Global recovery of a time-dependent coefficient for the wave equation from a single measurement , Asymptotic Analysis, 131:513-539, 2023

    2023

  23. [31]

    Gel’fand, I., Levitan, B., On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 309–360

    1951

  24. [32]

    the case of discrete spectrum , Transactions of the American Mathematical Society, 352:2765- 2787, 1999

    Gesztesy, F., Simon, B., Inverse spectral analysis with partial information on the potential, ii. the case of discrete spectrum , Transactions of the American Mathematical Society, 352:2765- 2787, 1999

    1999

  25. [33]

    C. S. Gordon, Survey of isospectral manifolds, Handbook of differential geometry 1, 747–778, 2000

    2000

  26. [34]

    S., Webb, D., Wolpert, S., Isospectral plane domains and surfaces via Riemannian orbifolds, Invent

    Gordon, C. S., Webb, D., Wolpert, S., Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math. 110:1, 1–22, 1992

    1992

  27. [35]

    Kluwer Academic Publishers, Dordrecht, 2001

    Craioveanu, M., Puta, M., Rassias, T., Old and new aspects in spectral geometry, Mathematics and its Applications, 534. Kluwer Academic Publishers, Dordrecht, 2001

    2001

  28. [36]

    An introduction., Cam- bridge University Press, Cambridge, 1994

    Grigis, A., Sj¨ ostrand, J.,Microlocal analysis for differential operators. An introduction., Cam- bridge University Press, Cambridge, 1994

    1994

  29. [37]

    Guillemin, V., Kazhdan, D., Some inverse spectral results for negatively curved 2-manifolds , Topology 19 (1980), 301–312

    1980

  30. [38]

    Haraux, A., S´ eries lacunaires et contrˆ ole semi-interne des vibrations d’une plaque rectangu- laire, J. Math. Pures Appl. (9) 68 (1989), no. 4, 457–465

    1989

  31. [39]

    Helin, T., Lassas, M., Oksanen, L., Inverse problem for the wave equation with a white noise source, Communications in Mathematical Physics, 332:933-953, 2013

    2013

  32. [40]

    Helin, T., Lassas, M., Oksanen, L., Saksala, T., Correlation based passive imaging with a white noise source , J. Math. Pures Appl. (9) 116 (2018), 132–160

    2018

  33. [41]

    Differential Equations 269 (2020), no

    Helin, T., Lassas, M., Ylinen, L., Zhang, Z., Inverse problems for heat equation and space-time fractional diffusion equation with one measurement , J. Differential Equations 269 (2020), no. 9, 7498–7528

    2020

  34. [42]

    Hezari, H., Zelditch, S., One can hear the shape of ellipses of small eccentricity , Ann. of Math. (2) 196, (2022), no 3, 1083–1136

    2022

  35. [43]

    Hochstadt, H., Lieberman, B., An inverse Sturm-Liouville problem with mixed given data , SIAM Journal on Applied Mathematics, 34:676-680, 1978

    1978

  36. [44]

    H¨ ormander, L.,The analysis of linear partial differential operators , vol. IV. Springer, Berlin (1983)

    1983

  37. [45]

    Nonlinear Differential Equations Appl., pages 179–219

    H¨ ormander, L.,On the uniqueness of the Cauchy problem under partial analyticity assump- tions, Geometrical optics and related topics (Cortona, 1996), volume 32 of Progr. Nonlinear Differential Equations Appl., pages 179–219. Birkh¨ auser Boston, Boston, MA, 1997

    1996

  38. [46]

    Imanuvilov, O., Yamamoto, M., Inverse parabolic problems by Carleman estimates with data taken at initial or final time moment of observation , Inverse Problems and Imaging, 2024, 18(2): 366-387

    2024

  39. [47]

    Jaffard, S., Contrˆ ole interne exact des vibrations d’une plaque rectangulaire, Portugal. Math. 47 (1990), no. 4, 423–429. 57

    1990

  40. [48]

    Isozaki, H., Some remarks on the multi-dimensional Borg-Levinson theorem , J. Math. Kyoto Univ., 31(1991), no. 3, 743–753

    1991

  41. [49]

    Kac, M., Can One Hear the Shape of a Drum? , The American Mathematical Monthly, 73(4P2), 1–23 (1966)

    1966

  42. [50]

    P., Sur les fonctions moyenne-periodiques bornees , Annales de l’Institut Fourier, 7:293–314, 1957

    Kahane, J. P., Sur les fonctions moyenne-periodiques bornees , Annales de l’Institut Fourier, 7:293–314, 1957

    1957

  43. [51]

    Partial Differential Equations 23 (1998), no

    Katchalov, A., Kurylev, Y., Multidimensional inverse problem with incomplete boundary spec- tral data, Comm. Partial Differential Equations 23 (1998), no. 1-2, 55–95

    1998

  44. [52]

    Katchalov, A., Kurylev, Y., Lassas, M., Inverse Boundary Spectral Problems , Chapman & Hall/CRC, 2001, 290 pp

    2001

  45. [53]

    2, 419–436

    Katchalov, A., Kurylev, Y., Lassas, M., Mandache, N., Equivalence of time-domain inverse problems and boundary spectral problems , Inverse Problems 20 (2004), no. 2, 419–436

    2004

  46. [54]

    Kian, Y., Li, Z., Liu, Y., Yamamoto, M., The uniqueness of inverse problems for a fractional equation with a single measurement , Mathematische Annalen, 380:1465-1495, 2020

    2020

  47. [55]

    Kian, Y., Liu, H., Uniqueness and stability in determining the wave equation from a single passive boundary measurement, preprint 2025, https://arxiv.org/abs/2507.10012

    2025 arXiv

  48. [56]

    Kian, Y., Uhlmann, G., Determination of the sound speed and initial source in photoacoustic tomography, Trans. Amer. Math. Soc. 378 (2025), no. 8, 5329–5353

    2025

  49. [57]

    Knox, C., Moradifam, A., Determining both the source of a wave and its speed in a medium from boundary measurements, Inverse Problems, 36(2):025002, 2020

    2020

  50. [58]

    Komornik, V., On the exact internal controllability of a Petrowsky system , J. Math. Pures Appl. 71 (1992), 331–342

    1992

  51. [59]

    Klibanov, M., Ioussoupova, O., Uniform Strict Convexity of a Cost Functional for Three- Dimensional Inverse Scattering Problem , SIAM J. Math. Anal., 26 (1995), pp. 147-179

    1995

  52. [60]

    Klibanov, M., Global convexity in a three-dimensional inverse acoustic problem , SIAM J. Math. Anal., 1997, 28, 1371-1388

    1997

  53. [61]

    Klibanov, M., Li, J., Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data , De Gruyter, 2021

    2021

  54. [62]

    Pures et Appl., 90 (2008), Issue 1, 42–59

    Krupchyk, K., Kurylev, Y., Lassas, M., Inverse Spectral Problems on a Closed Manifold , Journal Math. Pures et Appl., 90 (2008), Issue 1, 42–59

    2008

  55. [63]

    Differential Geom

    Kurylev, Y., Oksanen, L., Paternain, G., Inverse problems for the connection Laplacian , J. Differential Geom. 110, 457–494 (2018)

    2018

  56. [64]

    Kurylev, Y., Lassas, M., Yamaguchi, T., Uniqueness and stability in inverse spectral problems for collapsing manifolds , preprint 2012, https://arxiv.org/abs/1209.5875

    2012 arXiv

  57. [65]

    Laurent, C., L´ eautaud, M.,Lectures on unique continuation for waves , arXiv preprint (2023)

    2023

  58. [66]

    Laurent, C., L´ eautaud, M.,Uniform observability estimates for linear waves, ESAIM: Control, Optimisation and Calculus of Variations , 22 (4) (2016), 1097–1136

    2016

  59. [67]

    Lebeau, G., Contrˆ ole de l’´ equation de Schr¨ odinger, J. Math. Pures Appl. , 71 (1992), 267–291

    1992

  60. [68]

    Le Rousseau, J., Lebeau, G., Terpolilli, P., and Tr´ elat, E., Geometric control condition for the wave equation with a time-dependent observation domain , Anal. PDE Vol. 10 (2017), no. 4, 983–1015

    2017

  61. [69]

    Levitan, B., Gasymov, M., Determination of a differential equation by two of its spectra , Russian Mathematical Surveys, 19:1-63, 1964

    1964

  62. [70]

    Pure Appl

    Lin, F.-H., A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 43(1990), no. 1, 127–136

    1990

  63. [71]

    58 FEIZMOHAMMADI AND KRUPCHYK

    Liu, B., Quan, H., Saksala, T., Yan, L., H¨ older stability of an inverse spectral problem for the magnetic Schr¨ odinger operator on a simple manifold , preprint 2025, https://arxiv.org/abs/2507.13619v1. 58 FEIZMOHAMMADI AND KRUPCHYK

    2025 arXiv

  64. [72]

    Liu, H., Uhlmann, G., Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31(10):105005, 2015

    2015

  65. [73]

    Liu, S., Uhlmann, G., Yang, Y., An inverse hyperbolic problem with application to joint photoacoustic parameter determination, preprint 2024, https://arxiv.org/abs/2408.12759

    2024 arXiv

  66. [74]

    Lu, J., Stability of Gel’fand’s inverse interior spectral problem for Schr¨ odinger operators , preprint, https://arxiv.org/abs/2507.15560

    arXiv

  67. [75]

    https://doi.org/10.1038/nature10120

    Lundeen, J., Sutherland, B., Patel, A., Stewart, C., Bamber, Direct measurement of the quantum wavefunction , Nature 474, 188–191 (2011). https://doi.org/10.1038/nature10120

    2011 doi

  68. [76]

    Math., 49 (2021), 919– 936

    Maci` a, F.,Observability results related to fractional Schr¨ odinger operators, Vietnam J. Math., 49 (2021), 919– 936

    2021

  69. [77]

    Theory Adv

    Maci` a, F.,The Schr¨ odinger flow in a compact manifold: high-frequency dynamics and disper- sion, Modern aspects of the theory of partial differential equations, 275–289, Oper. Theory Adv. Appl., 216, Adv. Partial Differ. Equ. (Basel), Birkh¨ auser/Springer Basel AG, Basel, 2011

    2011

  70. [78]

    Milnor,J., Eigenvalues of the Laplace operator on certain manifolds , Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542

    1964

  71. [79]

    Murayama, R., The Gel’fand-Levitan theory and certain inverse problems for the parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 2, 317–330

    1981

  72. [80]

    Nachman, A., Sylvester, J., and Uhlmann, G., An n-dimensional Borg-Levinson theorem , Comm. Math. Phys., 115(1988), no. 4, 595–605

    1988

  73. [81]

    Novikov, R., A multidimensional inverse spectral problem for the equation −∆ψ + (v(x) − Eu(x))ψ = 0, Funct. Anal. Appl. 22 (1988), no. 4, 263–272 (1989)

    1988

  74. [82]

    Oksanen, L., Uhlmann, G., Photoacoustic and thermoacoustic tomography with an uncertain wave speed, Mathematical Research Letters, 21:1199-1214, 2013

    2013

  75. [83]

    Olevskii, A., Ulanovskii, A., Interpolation in Bernstein and Paley-Wiener spaces , Journal of Functional Analysis, 256:3257–3278, 2009

    2009

  76. [84]

    Pierce, A., Unique identification of eigenvalues and coefficients in a parabolic problem , SIAM Journal on Control and Optimization, 17:494-499, 1979

    1979

  77. [85]

    Rauch, J., Taylor, M., Exponential decay of solutions to hyperbolic equations in bounded do- mains, Indiana Univ. Math. J. 24 (1974), 79–86

    1974

  78. [86]

    Robbiano, L., Zuily, C., Uniqueness in the Cauchy problem for operators with partially holo- morphic coefficients, Invent. Math. 131 (1998), no. 3, 493–539

    1998

  79. [87]

    Saksala, T., Shedlock, A., An Inverse Problem for symmetric hyperbolic Par- tial Differential Operators on Complete Riemannian Manifolds , preprint 2025, https://arxiv.org/abs/2503.14676

    2025 arXiv

  80. [88]

    Seeley, R., Complex powers of an elliptic operator , Proc. Sympos. Pure Math., Amer. Math. Soc., 1967, 288–307

    1967

  81. [89]

    Andersson

    Shubin, M., Pseudodifferential operators and spectral theory, Translated from the 1978 Russian original by Stig I. Andersson. Second edition. Springer-Verlag, Berlin, 2001

    1978

  82. [90]

    Princeton University Press, Princeton, NJ, 2014

    Sogge, C., Hangzhou lectures on eigenfunctions of the Laplacian , Annals of Mathematics Studies, 188. Princeton University Press, Princeton, NJ, 2014

    2014

  83. [91]

    Stefanov, P., Uhlmann, G., Thermoacoustic tomography with variable sound speed , Inverse Problems, 25:075011, 2009

    2009

  84. [92]

    Stefanov, P., Uhlmann, G., Recovery of a source term or a speed with one measurement and applications, Transactions of the American Mathematical Society, 365:5737-5758, 2011

    2011

  85. [93]

    Suzuki, T., Inverse problems for heat equations on compact intervals and on circles, I , Journal of The Mathematical Society of Japan, 38:39-65, 1986. 59

    1986

  86. [94]

    Japan Acad

    Tanikawa, M., The spectrum of the Laplacian and smooth deformation of the Riemannian metric, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979) no. 7, 125–127

    1979

  87. [95]

    Partial Differential Equations 20 (1995), no

    Tataru, D., Unique continuation for solutions to PDE’s; between H¨ ormander’s theorem and Holmgren ’s theorem, Comm. Partial Differential Equations 20 (1995), no. 5-6, 855–884

    1995

  88. [96]

    Tataru, D., Unique continuation for operators with partially analytic coefficients , J. Math. Pures Appl. 78 (1999), 505–521

    1999

  89. [97]

    Applied Mathe- matical Sciences, 116

    Taylor, M., Partial Differential Equations I, Basic Theory , Second edition. Applied Mathe- matical Sciences, 116. Springer, New York, 2011

    2011

  90. [98]

    4, 1059–1078

    Uhlenbeck, K., Generic Properties of Eigenfunctions , American Journal of Mathematics, 98 (1976), no. 4, 1059–1078

    1976

  91. [99]

    Zelditch, S., The inverse spectral problem , With an appendix by Johannes Sj¨ ostrand and Maciej Zworski. Surv. Differ. Geom., IX, Surveys in differential geometry. Vol. IX, 401–467, Int. Press, Somerville, MA, 2004. A. Feizmohammadi, Department of Mathematics, University of T...

    2004

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