Gel'fand's Inverse Interior Spectral Problem

• Gel'fand's inverse interior spectral problem is the challenge of recovering a closed Riemannian manifold's geometry and potential from locally observed spectral data.
• It employs reconstruction mechanisms such as wave propagation, heat-kernel analysis, and distance function approximations to convert interior data into global geometric information.
• The approach bridges continuous and discrete models, yielding stability estimates with log-log dependence and addressing nonuniqueness in spectral determination.

Gel'fand's inverse interior spectral problem concerns the recovery of global geometry or operator coefficients from spectral information observed in the interior rather than from full global knowledge or boundary data alone. In a modern closed-manifold formulation, the data are

$\Big(U,\{\lambda_j,\phi_j|_U\}_{j=1}^\infty\Big),$

where $U\subset M$ is a nonempty open subset and $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$; the objective is to reconstruct the closed Riemannian manifold $(M,g)$ and the potential $q$ (Lu, 21 Jul 2025). Closely related formulations replace $\phi_j|_U$ by the heat kernel $p(x,y,t)$ on $V\times V\times \mathbb{R}_+$ for an open set $V\subset X$, or by values of normalized eigenfunctions at a fixed interior point in one-dimensional and integrable models (Honda et al., 16 Feb 2026, Liu et al., 2023).

1. Classical formulations and the meaning of “interior” spectral data

In the broad spectral-geometric literature, the inverse interior spectral problem asks whether spectral data determine the underlying object up to isometry. For a bounded domain $\Omega\subset \mathbb{R}^n$, the basic data may be

$U\subset M$0

and for a compact boundaryless manifold $U\subset M$1,

$U\subset M$2

In that classical form, the question is whether the spectrum determines $U\subset M$3 or $U\subset M$4 up to isometry (Datchev et al., 2011).

The explicitly interior version is more local. For compact manifolds of bounded geometry, one may know the metric on a small interior ball $U\subset M$5 together with the eigenvalues and eigenfunction restrictions

$U\subset M$6

called the interior spectral data, or its finite truncation, the finite interior spectral data (Bosi et al., 2017). In the Schrödinger setting on a closed manifold, one instead fixes an open set $U\subset M$7 and studies the inverse problem from $U\subset M$8 (Lu, 21 Jul 2025).

A heat-kernel formulation is equivalent in a strong sense. On a compact $U\subset M$9 space, the local data are the heat kernel $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$0 on $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$1, and the eigenfunction expansion

$(-\Delta_g+q)\phi_j=\lambda_j\phi_j$2

shows that local heat-kernel data recover local spectral data (Honda et al., 16 Feb 2026). In one-dimensional and integrable settings, the “interior” observation may be even more localized: for the Camassa–Holm multipeakon problem, the data are $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$3 at a fixed interior point $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$4 (Liu et al., 2023).

2. Uniqueness, rigidity, and the limits of spectral determination

A central theme is that eigenvalues alone are usually insufficient. Heat trace coefficients are spectral invariants: for bounded domains, $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$5 encodes area or volume and $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$6 encodes boundary length or area, while McKean–Singer showed that the Euler characteristic is also spectral. At the same time, heat invariants do not determine full geometry, and Gordon–Webb–Wolpert constructed isospectral but nonisometric plane domains. The survey literature therefore treats the general inverse problem as only partially true: there are strong positive results, but also genuine counterexamples and open problems (Datchev et al., 2011).

Positive rigidity results are concentrated in special geometric classes. A ball is spectrally characterized among all domains, and Tanno proved that if $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$7, then a compact $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$8-manifold with the same Laplace spectrum as the round sphere $(-\Delta_g+q)\phi_j=\lambda_j\phi_j$9 is isometric to $(M,g)$0. For analytic and symmetric domains, wave trace invariants associated with a single periodic billiard trajectory yield local inverse results, and analogous rigidity appears for analytic surfaces of revolution and negatively curved manifolds under appropriate dynamical conditions (Datchev et al., 2011).

Interior data can be much stronger than bare spectra, but they do not remove nonuniqueness automatically. In compact $(M,g)$1 spaces whose regular set is an open weighted $(M,g)$2-Riemannian manifold with locally Lipschitz density and weak convexity, the heat kernel on any open subset uniquely determines the whole space up to measure-preserving isometry (Honda et al., 16 Feb 2026). By contrast, in the Camassa–Holm multipeakon interior problem, uniqueness depends sharply on the data regime. If

$(M,g)$3

then $(M,g)$4 determine $(M,g)$5 uniquely only in the nondegenerate case $(M,g)$6 with no vanishing $(M,g)$7; if $(M,g)$8 or $(M,g)$9 but not both, there are two or four reconstructions depending on the sign pattern of the spectrum, and if some $q$0, infinitely many reconstructions may occur, parameterized by connected manifolds of positive dimension (Liu et al., 2023).

3. Reconstruction mechanisms: waves, controllability, and distance functions

The dominant reconstruction mechanisms are hyperbolic. For compact manifolds with bounded geometry, the interior-data reconstruction program uses quantitative unique continuation for the wave equation together with a geometric Boundary Control method. From approximate finite interior spectral data, one reconstructs approximate Fourier coefficients of truncated functions, computes approximate volumes of specially sliced subdomains, identifies admissible index families, and then builds piecewise constant approximations to the interior distance map. This produces a finite metric space $q$1 approximating $q$2, after which a geometric Whitney-type theorem yields a smooth Riemannian manifold close to the original one (Bosi et al., 2017).

In the Schrödinger case on closed manifolds, a related but more quantitative scheme is organized around interior distance functions

$q$3

on a fixed ball $q$4. Approximate spectral data determine a finite family of piecewise constant approximations to these functions, from which one extracts corresponding points $q$5, approximate pairwise distances, and hence a finite metric space close to $q$6. The same work reconstructs the potential through the first eigenfunction $q$7: since

$q$8

an approximation of $q$9 and $\phi_j|_U$0 gives a discrete approximation of $\phi_j|_U$1. The approximation of $\phi_j|_U$2 is carried out by a weighted graph Laplacian built on a finite net $\phi_j|_U$3 (Lu, 21 Jul 2025).

In the heat-kernel formulation on compact $\phi_j|_U$4 spaces, the reconstruction route is similarly dynamical. Local heat-kernel data first recover restricted spectral data, then solutions of forced wave equations with sources supported in the observation set, then approximate controllability, and finally the set of local distance functions

$\phi_j|_U$5

The metric is recovered from Varadhan asymptotics,

$\phi_j|_U$6

while short-time diagonal asymptotics recover the measure density on the regular set (Honda et al., 16 Feb 2026).

4. Quantitative stability

A major development is the passage from uniqueness to stability. For compact manifolds in the class $\phi_j|_U$7, approximate finite interior spectral data determine a finite metric space $\phi_j|_U$8 and then a smooth Riemannian manifold approximating $\phi_j|_U$9, with an explicit $p(x,y,t)$0-type stability estimate in Gromov–Hausdorff distance (Bosi et al., 2017).

For Schrödinger operators on closed manifolds in a bounded-geometry class

$p(x,y,t)$1

the stability theory is sharper and more geometric. If the first $p(x,y,t)$2 eigenvalues and eigenfunctions on $p(x,y,t)$3 are known approximately,

$p(x,y,t)$4

then one obtains both a finite geometric approximation of $p(x,y,t)$5 and a discrete approximation of the potential $p(x,y,t)$6 with uniform estimates. The resulting comparison between two admissible manifolds yields

$p(x,y,t)$7

together with

$p(x,y,t)$8

for an appropriate $p(x,y,t)$9-isometry $V\times V\times \mathbb{R}_+$0. The paper emphasizes that this double-logarithmic dependence is much weaker than Hölder stability but is considered optimal in general geometries for this type of problem (Lu, 21 Jul 2025).

A parallel stability theory exists for the heat-kernel version under Ricci bounds. For closed manifolds in

$V\times V\times \mathbb{R}_+$1

an $V\times V\times \mathbb{R}_+$2-approximation of the heat kernel on a ball implies that the manifolds are close in Gromov–Hausdorff distance and that the local comparison map extends to an “almost unique” global Gromov–Hausdorff approximation (Honda et al., 16 Feb 2026). Across both the interior spectral and heat-kernel settings, the weak modulus of continuity reflects the logarithmic losses inherent in quantitative unique continuation.

5. One-dimensional, integrable, and singular variants

The interior spectral paradigm has highly explicit realizations in one-dimensional and integrable systems. For global conservative multipeakon solutions of the Camassa–Holm equation, the generalized spectral problem involves discrete measures

$V\times V\times \mathbb{R}_+$3

and the interior spectral data at a fixed point $V\times V\times \mathbb{R}_+$4 are $V\times V\times \mathbb{R}_+$5. The key identity is

$V\times V\times \mathbb{R}_+$6

where

$V\times V\times \mathbb{R}_+$7

This expresses a diagonal Green-function-type quantity in terms of local spectral data. Half-line Weyl–Titchmarsh functions, continued fractions, and a Sturm-type oscillation theorem then control the splitting of poles between the right and left half-line data. The same analysis yields the trace formula

$V\times V\times \mathbb{R}_+$8

which reconstructs the Camassa–Holm field from normalized eigenfunctions (Liu et al., 2023).

For non-selfadjoint Dirac systems on a finite interval with non-integrable regular singularities at interior points $V\times V\times \mathbb{R}_+$9, the inverse problem includes the reconstruction of the regular potential $V\subset X$0, the singular coefficients $V\subset X$1, and the boundary parameters. The spectral data are $V\subset X$2, where $V\subset X$3 are eigenvalues and $V\subset X$4 are residues of the Weyl matrix. The scalar Weyl function satisfies

$V\subset X$5

and equality of $V\subset X$6 implies uniqueness up to the gauge rotation recorded in the paper. A spectral mappings method gives a constructive recovery procedure, with a main linear equation in the Banach space of bounded sequences and the explicit potential formula

$V\subset X$7

(Gorbunov et al., 2015).

Canonical Hamiltonian systems provide a further extension of Gel'fand-type interior spectral reconstruction. In this setting the inverse problem is to reconstruct a det-normalized Hamiltonian $V\subset X$8 from a spectral measure $V\subset X$9. For PW-sampling measures, the reconstruction is organized around truncated Toeplitz operators

$\Omega\subset \mathbb{R}^n$0

the reproducing kernel $\Omega\subset \mathbb{R}^n$1, and generalized Hilbert transforms. One recovers

$\Omega\subset \mathbb{R}^n$2

and then the off-diagonal entry from the generalized Hilbert transform of $\Omega\subset \mathbb{R}^n$3. Homogeneous and quasi-homogeneous spectral measures lead to explicit scaling laws, while power-law Hamiltonians yield Bessel-function models (Makarov et al., 2022, Makarov et al., 12 Sep 2025).

A closely related application appears in fractional diffusion. For the one-dimensional time-fractional diffusion equation, one interior trace $\Omega\subset \mathbb{R}^n$4 on a finite time interval determines simultaneously a spatial coefficient $\Omega\subset \mathbb{R}^n$5 and a Robin coefficient $\Omega\subset \mathbb{R}^n$6, provided $\Omega\subset \mathbb{R}^n$7 is already known on a subinterval. The proof converts the time-domain observation into a spectral identity, then into equality of Weyl $\Omega\subset \mathbb{R}^n$8-functions for a Sturm–Liouville problem, thereby reducing the coefficient inverse problem to a partial inverse spectral problem (Jing et al., 2024).

6. Discrete analogues and the broader research picture

A related discrete direction replaces interior-open-set data by boundary spectral data on finite weighted graphs. For the graph Laplacian on a finite weighted graph with boundary, the data are $\Omega\subset \mathbb{R}^n$9, and under a boundary adjacency condition together with the Two-Points Condition, the unknown interior vertices, edges, and weights are uniquely determined up to a boundary-preserving graph isomorphism. The proof uses a discrete wave equation, a wavefront lemma identifying first arrival times with graph distances, boundary distance functions, and a spectral characterization of initial data supported at single vertices. The framework applies to finite trees, finite square, hexagonal, and triangular lattices, graphite lattices, finite square ladders, and several perturbations, while the Kagome lattice is cited as a failure case (Blåsten et al., 2021).

A more constructive graph result assumes that the edge weights and boundary vertex weights are known and seeks the interior vertex weights from Neumann boundary spectral data. The method has two stages: reconstruction of the Neumann-to-Dirichlet map for the graph wave equation from spectral data, and reconstruction of the interior vertex weight from that map by a discrete boundary control method. A unique continuation theorem is proved for graphs satisfying a foliation-like topological condition, and the resulting algorithm is implemented and validated on several numerical examples (Li et al., 2024).

Taken together, these developments show that Gel'fand's inverse interior spectral problem is no longer a single narrowly defined question. It now encompasses closed smooth manifolds, compact metric-measure spaces with synthetic Ricci bounds, integrable systems, singular one-dimensional operators, canonical Hamiltonian systems, and discrete graph models. The recurrent structures are local spectral data, wave propagation, unique continuation, distance-function reconstruction, and Weyl- or Herglotz-type analytic objects. This suggests that the modern theory is best understood as a family of inverse problems in which solvability and stability are controlled by the geometry of the observation set, the regularity class of the underlying space or operator, and the extent to which the local spectral data resolve hidden symmetries or pole-splitting ambiguities.