Boundary Spectral Data in Inverse Problems

• Boundary spectral data (BSD) are defined as the collection of eigenvalues and boundary traces of eigenfunctions measured at the boundary.
• They are essential in inverse problems to uniquely determine interior coefficients and potentials using Dirichlet-to-Neumann map reconstructions.
• BSD theory extends to various operator settings including Schrödinger, Robin, and bi-harmonic operators, with proven stability and uniqueness estimates.

Boundary spectral data (BSD) is the boundary-facing spectral information attached to an operator and used as input for inverse problems. In the standard multidimensional Borg-Levinson setting, BSD consists of the eigenvalues of an elliptic operator together with boundary traces of the corresponding eigenfunctions, most commonly the normal derivatives on the boundary. For the Dirichlet Schrödinger operator on a bounded domain or on an admissible Riemannian manifold, the canonical form is

$$\mathrm{BSD}(q)=\{(\lambda_k,\partial_\nu \varphi_k|_{\partial\Omega})\}_{k\ge 1} \quad\text{or}\quad \{(\lambda_k(q),\psi_k(q)):k\ge 1\},\ \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M},$$

and the central question is whether these data determine the interior coefficients uniquely and stably (Choulli, 2019).

1. Definitions and principal variants

In the Dirichlet Schrödinger setting on a bounded domain $\Omega\subset \mathbb R^d$, $d\ge 2$, with

$$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$

BSD is given by the Dirichlet eigenvalues and the Neumann boundary data of the eigenfunctions,

$$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$

with eigenvalues repeated with multiplicity (Soccorsi, 2019). On a smooth compact Riemannian manifold $(\mathcal M,\mathfrak g)$ with boundary, the same structure appears for

$L(q)=-\Delta_{\mathfrak g}+q,$

where BSD is

$$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$

(Choulli, 2019).

The term also appears in related but non-identical forms for other operators. For the Schrödinger-Robin operator, the data are the eigenvalues and boundary traces of the eigenfunctions themselves,

$$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$

with Robin condition $\partial_\nu u+\alpha u=0$ on $\Omega\subset \mathbb R^d$0 (Choulli et al., 9 Jul 2025). For the perturbed bi-harmonic operator

$\Omega\subset \mathbb R^d$1

BSD is the sequence $\Omega\subset \mathbb R^d$2, where

$\Omega\subset \mathbb R^d$3

on $\Omega\subset \mathbb R^d$4 (Aroua et al., 2023). In fourth-order one-dimensional problems with distribution coefficients, the relevant spectral data are the eigenvalues and weight numbers, written as pairs $\Omega\subset \mathbb R^d$5 (Bondarenko, 2023).

Setting	Typical BSD	Boundary observable
Dirichlet Schrödinger	$\Omega\subset \mathbb R^d$6	normal derivative
Schrödinger-Robin	$\Omega\subset \mathbb R^d$7	boundary trace
Perturbed bi-harmonic	$\Omega\subset \mathbb R^d$8	higher-order boundary tuple

In the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data, the boundary object is formulated through a Dirichlet-to-Neumann map $\Omega\subset \mathbb R^d$9, and the paper explicitly interprets this map, or its linearization, as encoding the relevant boundary spectral information for the inverse problem (Jaiswal et al., 8 Apr 2026). This suggests that BSD is best understood as a class of boundary-accessible spectral observables rather than a single rigid data format.

2. Determination of coefficients from BSD

The foundational multidimensional Borg-Levinson statement is that BSD uniquely determines the potential. On admissible Riemannian manifolds, if $d\ge 2$0, $d\ge 2$1, and

$d\ge 2$2

then $d\ge 2$3 (Choulli, 2019). In bounded Euclidean domains of dimension $d\ge 2$4, the same uniqueness principle is stated as

$d\ge 2$5

for real-valued bounded potentials (Soccorsi, 2019).

A major refinement is that exact knowledge of the full sequence is not always necessary. If all but finitely many eigenpairs coincide, then uniqueness still holds. This is stated for simple manifolds in the form

$d\ge 2$6

(Choulli, 2019). In bounded Euclidean domains, Isozaki’s theorem asserts that if the BSD agree from some index $d\ge 2$7 on, then the potentials are equal (Soccorsi, 2019).

Partial boundary observation is also sufficient in some settings. For a bounded, connected, convex domain with $d\ge 2$8 boundary and any non-empty open subset $d\ge 2$9, the restricted data

$$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$0

uniquely determine a bounded potential $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$1 (Kian et al., 2017). The proof uses the Boundary Control method and is formulated specifically for arbitrarily small $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$2, without requiring smoothness of $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$3.

The same theme persists beyond Dirichlet realizations. For the Schrödinger-Robin operator in dimension $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$4, uniqueness is obtained from asymptotic or incomplete BSD: if $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$5 and $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$6 for all $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$7, or for all $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$8, then $$A_q=-\Delta+q,\qquad D(A_q)=H_0^1(\Omega)\cap H^2(\Omega),$$9 under the stated regularity and geometric assumptions (Choulli et al., 9 Jul 2025). For the spectral fractional Laplacian, equality of the linearized Dirichlet-to-Neumann maps for all boundary data implies equality of the potentials almost everywhere in $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$0 (Jaiswal et al., 8 Apr 2026).

A recurrent misconception is that the spectrum alone is the relevant datum in multidimensional inverse spectral theory. The literature summarized here instead treats the boundary traces as structurally essential. One-dimensional comparison sharpens the point: in one dimension the spectrum alone does not uniquely determine $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$1, whereas the pair of Dirichlet eigenvalues and boundary measurements does (Soccorsi, 2019).

3. Stability theory

Stability estimates quantify how perturbations in BSD propagate to perturbations in the recovered coefficient. For the Dirichlet Schrödinger operator on a bounded domain, a central result is the Hölder-type estimate for partial spectral data: $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$2 where

$$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$3

and, for $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$4,

$$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$5

The same paper proves that Hölder stability survives when finitely many eigenvalues or boundary traces are unknown, and also when the spectral data are only known asymptotically with errors of order $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$6 for sufficiently large $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$7 (Choulli et al., 2011).

For unbounded real-valued potentials $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$8, $$\mathrm{BSD}(q)=\{(\lambda_n,\partial_\nu\varphi_n)\}_{n\in\mathbb N},$$9, the stability norm weakens naturally to $(\mathcal M,\mathfrak g)$0. If

$(\mathcal M,\mathfrak g)$1

then

$(\mathcal M,\mathfrak g)$2

for the Dirichlet Laplacian with unbounded potential (Kian et al., 2022). The corresponding uniqueness theorem is formulated with asymptotic eigenvalue agreement and square-summable differences of normal derivatives (Bellassoued et al., 2022).

On admissible manifolds, BSD also yields logarithmic-type stability. If $(\mathcal M,\mathfrak g)$3 denotes the BSD distance

$(\mathcal M,\mathfrak g)$4

then one has

$(\mathcal M,\mathfrak g)$5

under the hypotheses of the corollary in the manifold setting (Choulli, 2019).

For geometric inverse problems, stability is again typically logarithmic. If $(\mathcal M,\mathfrak g)$6 and $(\mathcal M,\mathfrak g)$7 are admissible conformal metrics and $(\mathcal M,\mathfrak g)$8 is the weighted BSD discrepancy, then

$(\mathcal M,\mathfrak g)$9

for complete BSD, with related log-type estimates for partial BSD on a measurable subset $L(q)=-\Delta_{\mathfrak g}+q,$0 or an open subset $L(q)=-\Delta_{\mathfrak g}+q,$1 (Choulli et al., 2023).

Higher-order operators exhibit the same pattern. For the perturbed bi-harmonic operator, asymptotic BSD determines $L(q)=-\Delta_{\mathfrak g}+q,$2, and the paper states Hölder-type stability estimates for both $L(q)=-\Delta_{\mathfrak g}+q,$3 and $L(q)=-\Delta_{\mathfrak g}+q,$4 in $L(q)=-\Delta_{\mathfrak g}+q,$5 (Aroua et al., 2023). For the Schrödinger-Robin operator, asymptotic-full and asymptotic-incomplete BSD yield

$L(q)=-\Delta_{\mathfrak g}+q,$6

under the conditions stated in the two theorems (Choulli et al., 9 Jul 2025).

4. Analytical mechanisms: from BSD to Dirichlet-to-Neumann maps

A central analytic mechanism is the reconstruction or control of a Dirichlet-to-Neumann map from BSD. On admissible manifolds, the Dirichlet-to-Neumann map

$L(q)=-\Delta_{\mathfrak g}+q,$7

is represented in terms of BSD by

$L(q)=-\Delta_{\mathfrak g}+q,$8

and one has

$L(q)=-\Delta_{\mathfrak g}+q,$9

(Choulli, 2019). In this framework, uniqueness and stability follow from the chain

$$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$0

For multidimensional Borg-Levinson stability with partial data, the approach is described as using high-frequency analysis and explicit control over the Dirichlet-to-Neumann map via boundary spectral data, and explicitly not requiring wave equation or boundary control methods (Choulli et al., 2011). In Euclidean domains, the Dirichlet-to-Neumann map at complex frequencies is also the device through which BSD is analytically continued and related to the Fourier transform of $$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$1 (Soccorsi, 2019).

The relation becomes more systematic in inverse problems for higher-order coefficients. There the main methodological innovation is a precise connection between BSD and the family of elliptic and hyperbolic Dirichlet-to-Neumann maps. For the elliptic map $$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$2, BSD controls its derivatives at $$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$3, while for the hyperbolic map $$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$4 one has a direct norm estimate in terms of the BSD discrepancy $$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$5 (Choulli, 4 Oct 2025). This bridge transfers uniqueness and stability from inverse boundary value problems to inverse spectral problems.

In one-dimensional Sturm-Liouville theory, the corresponding object is the boundary data map $$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$6, defined by comparing two boundary trace maps associated with different self-adjoint extensions. It generalizes the spectral-parameter-dependent Dirichlet-to-Neumann map and enters Krein’s resolvent formula, trace formulas, and the spectral shift function. Its poles and determinants encode the spectrum and resolvent differences of self-adjoint extensions (Clark et al., 2012). A plausible implication is that BSD, Dirichlet-to-Neumann maps, and boundary data maps should be viewed as closely related realizations of the same boundary-response principle across operator classes.

5. Incomplete, asymptotic, and partial observations

A defining feature of modern BSD theory is that exact full data are often not necessary. One form of incompleteness is finite loss of eigenpairs. For the Dirichlet Schrödinger operator, if finitely many eigenvalues or normal derivative traces are unknown, the Hölder stability estimate remains valid for the remaining tail of the data, with constants depending on the number of missing terms (Choulli et al., 2011). The corresponding uniqueness statement for incomplete BSD is also explicit in the Euclidean and manifold settings (Soccorsi, 2019).

A second form is asymptotic knowledge. If, for all $$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$7,

$$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$8

and

$$\left\{(\lambda_k(q),\psi_k(q)):k\ge 1\right\},\qquad \psi_k(q)=\partial_\nu\phi_k|_{\partial\mathcal M}$$9

with $$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$0, then

$$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$1

(Choulli et al., 2011). In related work, asymptotic agreement of the eigenvalues together with square-summable differences of the boundary normal derivatives yields uniqueness for unbounded potentials (Bellassoued et al., 2022).

A third form is restriction of the observation set on the boundary. For any non-empty open $$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$2, the partial BSD

$$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$3

uniquely determines bounded $$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$4 in a convex domain (Kian et al., 2017). For Dirichlet-Laplace-Beltrami operators, partial BSD on a measurable subset $$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$5 or an open connected subset $$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$6 supports quantitative stability, provided the conformal factors agree in a neighborhood of the observed part of the boundary (Choulli et al., 2023).

The Robin setting displays a mixed version of incompleteness and asymptotics. The main theorem establishes Hölder stability from the asymptotic behavior of the eigenvalues together with the sequence of boundary measurements where finitely many terms are missing (Choulli et al., 9 Jul 2025). This body of results rules out the overly narrow view that BSD theory concerns only exact full sequences observed on the whole boundary.

6. Extensions, applications, and structural themes

BSD has expanded well beyond the classical Dirichlet Schrödinger problem. For higher-order coefficients of elliptic operators, BSD is used to determine metrics, conformal factors, conductivities, and potentials, with uniqueness and stability inequalities derived through the link to elliptic and hyperbolic Dirichlet-to-Neumann maps (Choulli, 4 Oct 2025). For the Dirichlet-Laplace-Beltrami operator, the data determine the operator only up to gauge equivalence, and in the conformal setting stability is obtained for the conformal factor rather than for an arbitrary representative of the metric class (Choulli et al., 2023).

The theory also feeds directly into time-dependent inverse problems. In parabolic inverse coefficient problems for

$$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$7

the strategy is to prove that the parabolic Dirichlet-to-Neumann map determines the BSD and then invoke multidimensional Borg-Levinson theory to recover $$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$8. The paper explicitly formulates this as a reduction of the parabolic inverse problem to a spectral identification problem (Soccorsi, 2019). This suggests a broader role for BSD as an intermediary between static spectral measurements and dynamic boundary observations.

In one-dimensional nonstandard Sturm-Liouville problems, the data can be markedly different. For a Sturm-Liouville operator with non-separated boundary conditions and a spectral parameter in one boundary condition, the inverse problem uses the spectrum of one boundary value problem together with a sequence of signs $$\mathrm{BSD}(q)=\{(\lambda_n,\psi_n):n\in\mathbb N\},\qquad \psi_n=\phi_n|_\Gamma,$$9, and the boundary value problem is uniquely determined by these data (Nabiev, 2019). In this setting BSD is effectively minimized relative to more classical formulations based on two spectra or norming constants.

Fourth-order problems provide another variation. For

$\partial_\nu u+\alpha u=0$0

sharp asymptotics are derived for eigenvalues and weight numbers under several separated boundary conditions, and these asymptotics are described as crucial for inverse spectral problems with regular or distributional coefficients (Bondarenko, 2023). For the perturbed bi-harmonic operator, asymptotic BSD controls first- and zero-order coefficients and extends multidimensional Borg-Levinson theory to a fourth-order setting (Aroua et al., 2023).

The spectral fractional Laplacian provides a nonlocal endpoint of the theory. There the Dirichlet-to-Neumann map for $\partial_\nu u+\alpha u=0$1 with inhomogeneous Dirichlet data is the relevant boundary object, and the paper also proves an additional density result that underpins inverse problems by ensuring sufficient richness of the associated solution space (Jaiswal et al., 8 Apr 2026). A plausible implication is that BSD has become a unifying language for boundary-based identification across local, nonlocal, second-order, and higher-order operators, even though the concrete boundary observables vary substantially from one class to another.