Improved stability of low Fourier modes in inverse problems for potentials

Abstract: We prove Lipschitz, sub-Hölder and Hölder stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider [ (-Δ- λ^2 + q)u=0 \quad \text{in} \quad Ω\subset \mathbb{R}^n. ] We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $λ$ is sufficiently large relative to $M<sup>{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider [ (-Δ+ q)u=0 \quad \text{in} \quad (-π,π)^n. ] We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-Hölder stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e<sup>{-c|k|<sup>{n/2}}$. We prove that the low Fourier modes can be recovered with Hölder stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L<sup>\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.

• The paper proposes methodologies and stability estimates for recovering low Fourier modes in elliptic inverse problems.
• Bandlimited potentials yield uniform Lipschitz stability at higher frequencies, while sub-Hölder and Hölder stability results are derived for real-analytic and super-exponentially decaying Fourier coefficients.
• The stability bounds hold under the condition that only the difference of the geometries satisfies regularity assumptions and hinge on key parameter conditions: $ ext{Q}=q-q'$ among others

Overview

This paper, by Liimatainen, Salo, and Trad (2608.20673), establishes stability estimates for recovering the low Fourier modes of an unknown potential from Dirichlet-to-Neumann (DN) map data in elliptic inverse problems. The central contribution is a set of three theorems covering different regularity regimes for the potential difference $Q = q - q'$: bandlimited (Lipschitz stability at large frequency $\lambda$), real-analytic periodic (sub-Hölder stability), and super-exponentially decaying Fourier coefficients (Hölder stability). A unifying feature across all three results is that only the difference $q - q'$ must satisfy the regularity assumption; the individual potentials are merely assumed to be bounded in $L^\infty$. This departs from much of the existing literature on Lipschitz stability, which typically requires the unknowns to lie in a fixed finite-dimensional subspace and yields constants growing exponentially with dimension.

The work is motivated by the "increasing stability" phenomenon initiated by Isakov, and more directly by recent work on increasing resolution showing that individual Fourier features can be recovered Lipschitz stably in linearized inverse scattering via singular value estimates (2608.20673). Extending such feature-level stability to nonlinear inverse problems is precisely what this paper accomplishes.

Bandlimited differences and Lipschitz stability

The first main result concerns the Helmholtz-type boundary value problem $(-\Delta - \lambda^2 + q)u = 0$ on $\Omega = (-\pi,\pi)^n$, $n \geq 2$, where $\lambda > 0$ is chosen so that the DN map $\Lambda_q(\lambda)$ is well-defined. If $q$ and $\lambda$0 are bounded by $\lambda$1 and their difference $\lambda$2 is $\lambda$3-bandlimited, then

$\lambda$4

whenever $\lambda$5, with $\lambda$6 independent of both $\lambda$7 and $\lambda$8. The Lipschitz constant is therefore uniform in the number of recovered modes, contrasting sharply with finite-dimensional results where the constant grows exponentially in dimension.

The proof proceeds through real geometrical optics solutions of the form $\lambda$9, constructed via a Neumann series argument using outgoing resolvent bounds for $q - q'$0 in weighted spaces. Substituting these solutions into the Alessandrini identity produces, for each Fourier index $q - q'$1, an estimate coupling the desired Fourier coefficient to the DN map difference plus remainder terms involving the errors $q - q'$2, $q - q'$3, and $q - q'$4. These remainders are controlled using Parseval's identity and Cauchy–Schwarz, yielding terms proportional to $q - q'$5 and $q - q'$6. On general domains $q - q'$7, the left-hand side becomes a Landau–Pollak–Slepian matrix $q - q'$8, which is shown to be positive definite and hence invertible; on the full torus cube, $q - q'$9 is the identity, giving the clean statement above.

A notable technical point is that all estimates are stated in the $L^\infty$0 operator norm rather than the natural $L^\infty$1 norm. The paper's appendix justifies this by proving that $L^\infty$2 even for merely bounded potentials, via a duality argument — a fact that is standard for smooth coefficients (where the difference of DN maps is order-zero pseudodifferential) but requires separate treatment here.

An important limitation of this result is the threshold condition $L^\infty$3: the frequency must grow polynomially with bandwidth for the estimate to hold. Moreover, on domains strictly smaller than the cube, the required constant involves $L^\infty$4, which grows as $L^\infty$5, degrading the uniformity of the stability bound.

Real-analytic differences and sub-Hölder stability

The second result addresses the case $L^\infty$6 with $L^\infty$7: if $L^\infty$8 is real-analytic, periodic, and extends holomorphically to a Grauert tube $L^\infty$9 of radius $(-\Delta - \lambda^2 + q)u = 0$0 with $(-\Delta - \lambda^2 + q)u = 0$1, then writing $(-\Delta - \lambda^2 + q)u = 0$2, there exist $(-\Delta - \lambda^2 + q)u = 0$3 and a function $(-\Delta - \lambda^2 + q)u = 0$4 as $(-\Delta - \lambda^2 + q)u = 0$5 such that

$(-\Delta - \lambda^2 + q)u = 0$6

This modulus is sub-Hölder: it decays slower than any power $(-\Delta - \lambda^2 + q)u = 0$7 but faster than any negative power of $(-\Delta - \lambda^2 + q)u = 0$8, interpolating between Hölder and logarithmic stability. The exponent $(-\Delta - \lambda^2 + q)u = 0$9 arises because the balancing of the two error contributions occurs at scale $\Omega = (-\pi,\pi)^n$0, reflecting the $\Omega = (-\pi,\pi)^n$1-power cost of the CGO construction relative to the exponential decay of analytic Fourier coefficients.

The proof combines complex geometric optics (CGO) solutions with phases $\Omega = (-\pi,\pi)^n$2, $\Omega = (-\pi,\pi)^n$3 with $\Omega = (-\pi,\pi)^n$4, inserted into the Alessandrini identity. The key structural step is choosing $\Omega = (-\pi,\pi)^n$5 with $\Omega = (-\pi,\pi)^n$6 small enough that $\Omega = (-\pi,\pi)^n$7, allowing the self-interaction term $\Omega = (-\pi,\pi)^n$8 on the right-hand side to be absorbed into the left-hand side. Tail sums over $\Omega = (-\pi,\pi)^n$9 are handled by a shell-decomposition lemma bounding $n \geq 2$0. Balancing the remaining two terms $n \geq 2$1 and $n \geq 2$2 determines $n \geq 2$3 uniquely, since $n \geq 2$4 is strictly increasing.

Two aspects deserve emphasis. First, the estimate exhibits explicit dependence on $n \geq 2$5: a larger domain of holomorphicity yields stronger stability, quantifying the intuitive principle that smoother unknowns are more stably recoverable. Second, since $n \geq 2$6, measurement precision directly translates into resolution — a quantitative resolution–noise tradeoff in the fully nonlinear problem. The authors note a genuine restriction: periodicity of $n \geq 2$7 is essential for the integration-by-parts argument yielding exponential Fourier decay. Without it, boundary terms of the form $n \geq 2$8 cannot be simultaneously suppressed by a single choice of $n \geq 2$9, preventing the exponential decay estimate. The authors suggest that boundary determination results for analytic potentials could remedy this, but leave this open.

Super-exponential Fourier decay and Hölder stability

The third result strengthens the coefficient decay assumption to $\lambda > 0$0 for some $\lambda > 0$1 — faster than the exponential decay characteristic of real-analytic functions when $\lambda > 0$2. Under this hypothesis, the same CGO framework delivers genuine Hölder stability:

$\lambda > 0$3

The Hölder exponent depends explicitly and monotonically on the decay rate $\lambda > 0$4, approaching 1 (near-Lipschitz) as $\lambda > 0$5. The mechanism is transparent from the proof structure: the stretched tail bound reduces $\lambda > 0$6 to $\lambda > 0$7, matching the scale of the amplification term $\lambda > 0$8, so both balance exactly at $\lambda > 0$9. This upgrades sub-Hölder to Hölder purely through the improved tail control, illustrating how the regularity class of the unknown directly dictates the achievable modulus of continuity for its low-frequency content.

As a corollary remarked upon by the authors, if $\Lambda_q(\lambda)$0 is instead $\Lambda_q(\lambda)$1-bandlimited with fixed $\Lambda_q(\lambda)$2, the tail vanishes identically and one recovers Lipschitz stability $\Lambda_q(\lambda)$3 without any smallness assumption. The authors compare this to prior finite-measurement results, noting their constant grows as $\Lambda_q(\lambda)$4 rather than $\Lambda_q(\lambda)$5 — a worse dimensional dependence — in exchange for requiring only the difference $\Lambda_q(\lambda)$6 to be bandlimited while the individual potentials remain unrestricted.

Limitations and open questions

Several restrictions qualify these results. All non-bandlimited theorems require $\Lambda_q(\lambda)$7 and assume $\Lambda_q(\lambda)$8 is not a Dirichlet eigenvalue of $\Lambda_q(\lambda)$9. The analytic case additionally assumes $q$0 and periodicity of the difference; extension to nonzero frequencies or non-periodic analytic differences remains unresolved. On general domains, the Lipschitz constant acquires a factor $q$1 that deteriorates with bandwidth, so the clean uniform-in-$q$2 behavior is specific to the cube. The dimensional scaling of the constants ($q$3 versus $q$4 in comparable finite-dimensional results) is not optimized, and whether the gap between the sub-Hölder modulus of the analytic case and the Hölder modulus of the stretched-exponential case is sharp — or whether intermediate decay rates yield intermediate moduli in a precise sense — is not addressed.

Conclusion

This paper demonstrates that in nonlinear inverse problems for Schrödinger potentials, stable recoverability of individual Fourier features persists beyond the linearized regime, provided the difference of the potentials lies in an appropriate regularity class. The progression from Lipschitz (bandlimited, large $q$5) through sub-Hölder (real-analytic) to Hölder (super-exponential decay) stability, each with a number of recoverable modes increasing with data quality or frequency and with constants uniform in that number, gives a quantitative account of how regularity and measurement precision jointly determine resolution. The requirement that only $q$6 satisfy the structural assumptions, with $q$7 merely bounded, substantially weakens the hypotheses needed for Lipschitz-type guarantees relative to the finite-dimensional literature.