Quantitative Borg-Levinson Theorem

• Quantitative Borg-Levinson theorem is a set of inverse spectral results that guarantees unique recovery of potentials or magnetic fields with explicit Hölder or logarithmic stability estimates.
• It utilizes methodologies like complex geometric optics, resolvent estimates, and Weyl laws to link boundary observables to interior coefficients.
• The framework adapts to various elliptic operators and boundary conditions, making it applicable to unbounded potentials and manifold settings.

The quantitative Borg-Levinson theorem refers to a family of inverse spectral results establishing not only uniqueness but also explicit stability estimates for the recovery of potential terms or magnetic fields in partial differential operators (notably Schrödinger, magnetic Schrödinger, biharmonic, and Robin Laplacians), from boundary spectral data. Rather than only guaranteeing unique determination (classical Borg-Levinson), these quantitative results provide Hölder-type or logarithmic-type moduli of continuity in norms of the potential, controlled by explicit metrics on the discrepancy of spectral data. Modern developments encompass unbounded potentials, partial or asymptotic spectral data, and extension to Riemannian manifolds and higher-order operators.

1. Multi-dimensional Borg-Levinson Setup and Spectral Data

The generic setting involves a bounded domain $\Omega\subset\mathbb{R}^n$ (or a smooth compact manifold $(M,g)$), with self-adjoint operators such as the Schrödinger operator $-\Delta+q$ or its magnetic, biharmonic, or Robin-perturbed analogs. Boundary spectral data (BSD) incorporates:

• The discrete spectrum $\{ \lambda_k \}_{k=1}^{\infty }$ of eigenvalues (counted with multiplicity).
• Boundary observables such as the normal derivative traces $\partial_\nu\phi_k|_{\partial\Omega}$ (Dirichlet case), magnetic normal traces $(\nabla_A\phi_k)\cdot\nu|_{\Gamma}$ (magnetic case), or Dirichlet traces for Neumann realizations.
• For higher-order (biharmonic) operators, two families of traces are needed: $\partial_\nu\varphi_{k}$ and $\partial_\nu(\Delta\varphi_k)$.

The potential $q$ or pair $(A,q)$ is sought in a suitable function space (e.g., $L^\infty$, $H^m$, $L^{n/2}$, or $W^{1,\infty}$), potentially without boundedness, enabling a wider class of admissible coefficients (Choulli et al., 2011, Choulli, 2024).

Partial or asymptotic data are accommodated by allowing finitely many spectral modes to be unknown, or more generally, error terms $O(k^{-a})$ for tail behavior of the spectral closeness.

2. Quantitative Hölder and Logarithmic Stability Estimates

Central quantitative results establish explicit stability bounds of Hölder-type (and logarithmic-type in certain partial data scenarios):

• Dirichlet Schrödinger (full or partial data): If the spectral data of $q_1,q_2$ above an index $K$ are close, with $d(K)$ quantifying $\ell^2$ differences of eigenvalues and boundary traces, then

$\|q_1-q_2\|_{L^2(\Omega)} \le C\, d(K)^\alpha$

with constants $C,\alpha$ depending on regularity, domain, and bounds (not $K$) (Choulli et al., 2011).

• Asymptotic data: If eigenvalue and boundary trace discrepancies decay at rates $k^{-a}$, then

$$\|q_1-q_2\|_{L^2(\Omega)} \le C_a\, \epsilon^{\alpha(a)}$$

for $\epsilon$ the "level error", with convergence to exact recovery as $a$ increases (Choulli et al., 2011).

• Unbounded potentials and magnetic extensions: For $V_1,V_2 \in L^{n/2}(\Omega)$, define

$$D_e(V_1,V_2) = \sum_{k=1}^\infty k^{-2/n} (|\lambda_k^{(1)}-\lambda_k^{(2)}| + \|\partial_\nu \phi_k^{(1)} - \partial_\nu \phi_k^{(2)}\|_{L^2(\partial\Omega)})$$

then

$$\|V_1 - V_2\|_{H^{-1}(\Omega)} \le C\, D_e(V_1,V_2)^{1/(n+2)}$$

(Choulli, 2024).

• Magnetic Schrödinger and Robin with unbounded potential: The stability exponent and norm are adjusted for function space, operator, and boundary condition:
  • For magnetic field recovery (under curl constraints): $$\| \operatorname{curl}(A_1-A_2) \|_{L^2(\Omega)} \le c_+ [\delta(b_1,b_2)]^{\beta_1}$$ (Choulli et al., 22 Jan 2026).
  • Robin Laplacian: $$\|q-\tilde q\|_{H^{-1}(\Omega)}\le C \left\{ \sum |\lambda_k-\tilde\lambda_k| + \sum \| U_k - \tilde U_k \|^{2} \right\}^{\alpha}$$ for full data, and $C\psi(\cdots)$ for partial data with logarithmic modulus (Choulli et al., 2022).
• Biharmonic operator: For $\Delta^2+q$ with Navier boundary conditions, knowledge of all but finitely many eigenvalues and two boundary traces yields

$\|q_1-q_2\|_{L^2(B_R)} \le C\, \varepsilon^\theta$

with $\varepsilon$ incorporating all mode discrepancies, and $\theta$ explicit in $n,m$ (Li et al., 2021).

3. Analytical and Methodological Foundations

Quantitative Borg-Levinson theorems exploit several rigorous analytic structures:

• Spectral Representation: DtN maps admit meromorphic expansions in the spectral parameter, directly expressing them in terms of BSD.
• CGO (Complex Geometric Optics) Construction: High-frequency solutions (either plane waves in Euclidean domains or geodesic wave packets in manifolds) allow reduction to Fourier or ray transforms linking the spectral error to the norm of the unknown coefficient.
• Resolvent and Weyl Law Estimates: Resolvent bounds in $L^p$ and Sobolev spaces, together with Weyl asymptotics (eigenvalue growth $k^{2/n},\,k^{4/n}$) are essential for quantifying high-frequency tails and for controlling partial or asymptotic data.
• Optimization: The choice of large spectral parameters is optimized to balance the main term against the decaying tail, yielding the stability exponent.
• Gauge-Invariant Formulation: For magnetic potentials $A$, only the solenoidal part $A^s$ (co-closed form) is reconstructible (Bellassoued et al., 2018).

In the Robin and biharmonic cases, variational techniques are adapted. Unique continuation properties and interpolation inequalities support stability under partial boundary observation.

4. Classes of Potentials and Domains: Extensions and Limitations

• Unbounded Potentials: In domains of dimension $n\geq 5$ (or lower with integrability adjustment), potentials in $L^{n/2}$ (not necessarily $L^\infty$) are stably reconstructible in the $H^{-1}$-norm (Choulli, 2024, Choulli et al., 2022).
• Boundary Conditions: Dirichlet results extend directly; Robin boundary cases require careful control of the boundary coefficient $a\in L^s(\partial\Omega)$, and in the partial data scenario necessitate $q=\tilde{q}$ in a neighborhood of the boundary (Choulli et al., 2022).
• Manifold Settings: On simple compact Riemannian manifolds with strictly convex boundaries, the theorems persist with adjustments for ray transforms and pseudo-differential operator theory (Bellassoued et al., 2018, Choulli, 2024).

A plausible implication is that the robustness of the method permits adaptation to more general elliptic operators, provided appropriate control on the spectral asymptotics and boundary trace regularity.

5. Dependence of Stability Exponents, Constants, and Spectral Metrics

All constants and exponents in the stability bounds depend only on geometric properties (dimension $n$, domain smoothness), a priori bounds on potentials or magnetic fields, and integrability indices:

Parameter	Influence on Constants/Exponent	Typical Values
Dimension $n$	Lower exponent for higher $n$	$\alpha = 1/(n+2)$ for unbounded case
Integrability $r$	Larger $r$ gives higher $\alpha$	$\alpha=(1-2\beta)/(3(n+2))$, $\beta=n(2-r)/(2r)$
Bound on potential	Required for finite constant	$L^\infty$ or envelope $V_0\in L^{n/2}$
Spectral data discrepancy	Determines modulus argument	$\ell^2$ for traces, $\ell^1$ or rates for eigenvalues

For partial boundary data, only logarithmic-type stability is generally available, which is sharp in view of unique continuation phenomena (Choulli et al., 2022). In magnetic cases, stability in the $L^2$-norm is available for the electric potential; for the magnetic field only the solenoidal part is controlled, in accordance with gauge invariance (Bellassoued et al., 2018).

6. Implications, Extensions, and Technical Challenges

The quantitative Borg-Levinson framework yields explicit estimates vital for practical inverse problems where spectral data may be incomplete, noisy, or only asymptotically accurate. It demonstrates a rigorous link between spectral measurement errors and reconstruction fidelity, under broad admissibility classes for coefficients. The proof strategies are sufficiently flexible to treat unbounded potentials, non-standard boundary conditions, and higher-order or magnetic operators, provided the analytic infrastructure (resolvent estimates, Weyl laws) is available (Choulli et al., 2011, Choulli, 2024, Choulli et al., 22 Jan 2026, Li et al., 2021).

A plausible implication is that continued technical improvements in resolvent theory and trace inequalities may extend these results to even broader operator classes (e.g., variable coefficients, systems), though fundamental limits appear in the rate (Hölder or logarithmic) determined by the eigenvalue growth and data regularity.

Recent advances also confirm the optimality of the modulus of continuity for partial data and reveal the natural role of the $H^{-1}$-norm for unbounded potential recovery (Choulli, 2024, Choulli et al., 2022). For manifolds, injectivity and stability ride on the structure of geodesic ray transforms, with technical adaptation for complex geometric optics on Riemannian structures (Bellassoued et al., 2018).

The quantitative Borg-Levinson theorem thus encompasses a spectrum of inverse spectral results, providing explicit stability for a wide class of elliptic operators under realistic data availability constraints.