Boundary Spectral Data Overview

• Boundary spectral data is the collection of eigenvalues and boundary measurements of eigenfunctions, encapsulating the operator’s response at the domain boundary.
• It serves as the discrete counterpart to continuous boundary response operators and is key in inverse spectral theory for deducing potentials, geometries, and coefficients.
• Methodologies such as Dirichlet-to-Neumann maps, boundary control, and spectral interpolation underlie the stability and uniqueness results achieved using BSD.

Searching arXiv for recent and foundational papers on boundary spectral data to ground the article. Boundary spectral data (BSD) denotes the spectral information attached to the boundary behavior of eigenfunctions. In a standard multidimensional Dirichlet Schrödinger setting, with $A_q=-\Delta+q$ on a bounded domain $\Omega\subset\mathbb R^n$, it takes the form

$$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$

where $\gamma\subset\partial\Omega$ is open, $\lambda_k$ are Dirichlet eigenvalues, and $\partial_\nu\phi_k$ are outward normal derivatives of the corresponding eigenfunctions (Kian et al., 2017). Closely related variants occur for Laplace–Beltrami operators, Robin problems, bi-harmonic operators, Steklov problems, and Sturm–Liouville systems, where the boundary observable may be a metric normal derivative, a boundary trace, a Neumann-type pair, or a boundary data map rather than $\partial_\nu\phi_k$ itself [(Choulli, 4 Oct 2025); (Choulli et al., 9 Jul 2025); (Aroua et al., 2023); (Imeri et al., 2022); (Clark et al., 2012)]. Taken together, these works indicate that BSD is the discrete spectral counterpart of boundary response operators and a central object in inverse spectral theory.

1. Core definitions and operator-dependent realizations

For the Dirichlet realization of the Schrödinger operator on a bounded $C^2$ domain,

$$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$

the spectrum is discrete and there exists an orthonormal basis of eigenfunctions $\{\phi_k\}\subset H_0^1(\Omega)\cap H^2(\Omega)$. Because $\Omega\subset\mathbb R^n$0 under the Dirichlet condition, the boundary value itself is trivial; the informative boundary quantity is the flux $\Omega\subset\mathbb R^n$1. This is why BSD is recorded as eigenvalues together with normal derivatives rather than boundary traces of the eigenfunctions (Kian et al., 2017).

For Laplace–Beltrami operators $\Omega\subset\mathbb R^n$2 on a bounded smooth domain, the corresponding data are

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

with $\Omega\subset\mathbb R^n$4 defined using the metric $\Omega\subset\mathbb R^n$5. The same pattern extends to conductivity operators $\Omega\subset\mathbb R^n$6, with $\Omega\subset\mathbb R^n$7, and to Schrödinger operators with potential $\Omega\subset\mathbb R^n$8, where BSD again records eigenvalues and boundary normal derivatives (Choulli, 4 Oct 2025).

For Robin boundary conditions, the role of the boundary observable changes. In the Schrödinger–Robin setting,

$\Omega\subset\mathbb R^n$9

the paper defines

$$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$0

Here the boundary trace itself is used, rather than $$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$1, because the boundary condition already couples the normal derivative to the trace (Choulli et al., 9 Jul 2025).

Higher-order operators require higher-order boundary traces. For the perturbed bi-harmonic operator

$$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$2

with Dirichlet-type conditions $$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$3 and $$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$4, the BSD is

$$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$5

where

$$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$6

This is the direct fourth-order analogue of the second-order pair $$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$7 (Aroua et al., 2023).

2. Relation to Dirichlet-to-Neumann maps and boundary operators

A recurring structural fact is that BSD encodes Dirichlet-to-Neumann operators. For Laplace–Beltrami operators, the elliptic boundary value problem

$$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$8

admits the spectral representation

$$\mathrm{BSD}(q,\gamma):=\{(\lambda_k,(\partial_\nu\phi_k)|_\gamma):k\in\mathbb N^*\},$$9

and, for sufficiently high derivatives,

$\gamma\subset\partial\Omega$0

In this sense, the eigenvalues are the poles and the boundary amplitudes are the residues of the elliptic DtN map. The same paper further shows that BSD also determines the hyperbolic DtN map for the associated wave equation (Choulli, 4 Oct 2025).

The boundary control method makes the same relation explicit in the time domain. For the wave equation with boundary source $\gamma\subset\partial\Omega$1 supported on $\gamma\subset\partial\Omega$2, if

$\gamma\subset\partial\Omega$3

then

$\gamma\subset\partial\Omega$4

Hence knowledge of $\gamma\subset\partial\Omega$5 determines the modal coefficients of all waves generated from $\gamma\subset\partial\Omega$6, and therefore determines global and localized inner products of wave fields (Kian et al., 2017).

In one-dimensional Sturm–Liouville theory, the analogous objects are the boundary data maps

$\gamma\subset\partial\Omega$7

which generalize spectral-parameter-dependent Dirichlet-to-Neumann maps. They satisfy group and inverse properties, enter Krein’s resolvent formula, and their determinants encode trace formulas and spectral shift functions (Clark et al., 2012). This suggests that BSD may be viewed either as a sequence of discrete boundary traces or as the meromorphic boundary operator whose poles and residues reproduce that sequence.

3. Uniqueness in inverse spectral problems

A central inverse question is whether BSD determines the underlying coefficients or geometry. For bounded real-valued $\gamma\subset\partial\Omega$8 on a bounded, connected, convex $\gamma\subset\partial\Omega$9 domain, equality of partial data

$\lambda_k$0

on an arbitrary nonempty open subset $\lambda_k$1 implies

$\lambda_k$2

The result is notable because $\lambda_k$3 may be arbitrarily small and because the potential is assumed only bounded, not smooth (Kian et al., 2017).

Uniqueness also persists in low-regularity asymptotic regimes. For real-valued electric potentials $\lambda_k$4, $\lambda_k$5, if

$\lambda_k$6

then $\lambda_k$7 a.e. in $\lambda_k$8. Here $\lambda_k$9, so uniqueness follows from asymptotic boundary spectral data rather than exact full data (Bellassoued et al., 2022).

For anisotropic geometric problems, the determination is modulo natural invariances. If two metrics $\partial_\nu\phi_k$0 have identical full BSD,

$\partial_\nu\phi_k$1

then there exists a $\partial_\nu\phi_k$2 diffeomorphism $\partial_\nu\phi_k$3 with $\partial_\nu\phi_k$4 such that

$\partial_\nu\phi_k$5

The same work also gives uniqueness for conformal factors, conductivities, and potentials in the corresponding admissible classes (Choulli, 4 Oct 2025).

In one dimension, partial boundary spectral information can be enough. For $\partial_\nu\phi_k$6 on $\partial_\nu\phi_k$7 with Dirichlet boundary conditions and normalization $\partial_\nu\phi_k$8, if a subset $\partial_\nu\phi_k$9 has density at least $\partial_\nu\phi_k$0, if

$\partial_\nu\phi_k$1

and if

$\partial_\nu\phi_k$2

then $\partial_\nu\phi_k$3 on $\partial_\nu\phi_k$4 (Feizmohammadi et al., 4 Sep 2025).

4. Partial, incomplete, and asymptotic data; stability theory

Beyond uniqueness, several works establish quantitative stability from incomplete or asymptotic BSD. For bounded potentials $\partial_\nu\phi_k$5 with $\partial_\nu\phi_k$6, one paper proves a Hölder estimate

$\partial_\nu\phi_k$7

where $\partial_\nu\phi_k$8 is built from the tail of the eigenvalue sequence and weighted differences of boundary normal derivatives. The same work allows finitely many eigenvalues and normal derivatives to be unknown, and also treats asymptotically known data with errors of order $\partial_\nu\phi_k$9, $C^2$0 large enough (Choulli et al., 2011).

For unbounded potentials, the stability problem persists in weaker norms. One paper considers $C^2$1, $C^2$2, assumes a square-summable sequence of differences of boundary normal derivatives, and proves that $C^2$3 can be Hölder stably retrieved through knowledge of the asymptotics of the eigenvalues (Kian et al., 2022). A closely related Robin result shows that an unbounded real-valued potential can be Hölder stably retrieved from the asymptotic behavior of the eigenvalues and the sequence of boundary measurements of the corresponding eigenfunctions where finitely many terms are missing (Choulli et al., 9 Jul 2025).

In geometric settings, the stability problem takes logarithmic or Hölder–logarithmic form. Stability inequalities for determining a Dirichlet–Laplace–Beltrami operator from complete and partial BSD are established on compact manifolds with boundary (Choulli et al., 2023). For higher-order coefficients, full BSD yields uniqueness and stability inequalities for anisotropic metrics, conformal factors, conductivities, and potentials; the moduli are logarithmic or Hölder-type depending on the class of coefficients and the ambient geometry (Choulli, 4 Oct 2025).

For higher-order perturbations of the bi-harmonic operator, asymptotic closeness of eigenvalues together with rescaled closeness of the Neumann-type traces implies $C^2$4 and yields Hölder stability for both the electric potential and the first-order coefficient in $C^2$5. This suggests that high-frequency BSD remains decisive even when low modes are uncontrolled (Aroua et al., 2023).

5. Methodological frameworks

Two methodological patterns dominate the literature. The first is the boundary control route. In the partial-data Schrödinger problem, the proof proceeds from equality of BSD to equality of modal wave coefficients, then to equality of global and localized inner products, then to pointwise identities of wave fields in a collar neighborhood $C^2$6, and finally to local and then global equality of the potential. The analytic ingredients are spectral expansions, finite speed of propagation, unique continuation, and approximate controllability (Kian et al., 2017).

The second is the BSD-to-DtN-to-coefficients route. In the higher-order coefficient problem, spectral expansions give explicit formulas for elliptic and hyperbolic DtN maps in terms of $C^2$7; analyticity in $C^2$8, large-$C^2$9 decay, and Taylor expansion of $$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$0 then yield DtN stability estimates; these are composed with Calderón-type or hyperbolic inverse results to recover the coefficients (Choulli, 4 Oct 2025). Taken together, these arguments present BSD as a compressed spectral encoding of boundary operators.

Isozaki-type representation formulas provide a third, complementary mechanism. For singular or unbounded potentials, one constructs special oscillatory solutions, defines boundary pairings such as $$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$1, and proves that

$$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$2

The spectral representation of the same boundary pairings in terms of BSD then converts asymptotic spectral control into vanishing of the Fourier transform of the coefficient difference (Bellassoued et al., 2022). Related oscillatory constructions also underlie the asymptotic Robin and bi-harmonic results (Choulli et al., 9 Jul 2025, Aroua et al., 2023).

In one dimension, finite-time boundary measurements can be converted into partial BSD by interpolation. Using Kahane’s interpolation theorem, endpoint traces of wave or Schrödinger evolutions are spectrally filtered so that one isolates individual frequencies and recovers large-index pairs $$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$3 from finite-time data. This replaces analyticity or infinite-time arguments by a density-based spectral interpolation mechanism (Feizmohammadi et al., 4 Sep 2025).

6. Related notions and broadened uses of boundary spectral data

The term also covers boundary-weighted spectral quantities that are not identical to the classical Borg–Levinson data. For Neumann eigenfunctions of the Laplacian, the natural boundary datum is not the raw trace $$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$4 but the weighted trace

$$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$5

where the spectral weight regularizes whispering-gallery concentration near glancing. The resulting family has uniform upper and lower $$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$6 bounds and satisfies quasi-orthogonality in spectral windows; this yields sharp inclusion bounds for Neumann eigenvalues and improved numerical methods of particular solutions (Barnett et al., 2015).

Steklov theory provides a particularly direct boundary-spectral model. The Steklov eigenpairs $$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$7 satisfy

$$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$8

so they are exactly the eigenpairs of the Dirichlet-to-Neumann map $$A_q=-\Delta_x+q(x),\qquad \mathcal D(A_q)=\{u\in H_0^1(\Omega): -\Delta u+qu\in L^2(\Omega)\},$$9. In this setting, $\{\phi_k\}\subset H_0^1(\Omega)\cap H^2(\Omega)$0 is itself the boundary spectral data, and it can be used constructively to approximate solutions of Dirichlet and Robin boundary value problems by Steklov spectral expansions (Imeri et al., 2022).

In one-dimensional problems with eigenvalue-dependent boundary conditions, the relevant boundary spectral object may be encoded by spectral identities rather than direct traces. For Schrödinger operators on $\{\phi_k\}\subset H_0^1(\Omega)\cap H^2(\Omega)$1 with rational Herglotz–Nevanlinna boundary coefficients, the paper derives a finite system of identities relating coefficients of a polynomial $\{\phi_k\}\subset H_0^1(\Omega)\cap H^2(\Omega)$2 attached to the boundary condition to renormalized moments of the eigenvalues and norming constants. These identities function as a boundary-coefficient analogue of the Gelfand–Levitan equation (Guliyev, 2019).

For higher-order ordinary differential operators with distribution coefficients and separated boundary conditions, the spectral data is taken to be the eigenvalues $\{\phi_k\}\subset H_0^1(\Omega)\cap H^2(\Omega)$3 together with weight numbers

$\{\phi_k\}\subset H_0^1(\Omega)\cap H^2(\Omega)$4

The resulting asymptotic formulas and comparison estimates quantify how distribution coefficients and boundary parameters enter the high-frequency spectral data (Bondarenko, 2022). This suggests that the notion of BSD extends naturally from PDE boundary traces to residue-type spectral invariants whenever the boundary conditions are encoded in characteristic determinants.

Taken together, these developments portray boundary spectral data as a family of operator-dependent boundary observables—sometimes traces, sometimes weighted traces, sometimes generalized boundary maps, sometimes norming residues—that mediate between discrete spectra and boundary response. Across elliptic, hyperbolic, geometric, higher-order, and one-dimensional settings, BSD functions as a boundary-facing spectral fingerprint of the underlying operator and, in many regimes, of the coefficients or geometry themselves.