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On Fractional Borg-Levinson Problem

This paper establishes the unique determination of potentials within a specific admissible class for the fractional Borg-Levinson inverse spectral problem, provided the fractional exponent lies in the interval $(\frac{1}{2}, 1)$ .

Original authors: Saumyajit Das (Harish Chandra Research Institute India), Tuhin Ghosh (Harish Chandra Research Institute India)

Published 2026-09-11

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CC BY 4.0

Original authors: Saumyajit Das (Harish Chandra Research Institute India), Tuhin Ghosh (Harish Chandra Research Institute India)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of mathematics and physics, there exists a class of problems known as inverse spectral problems. Imagine a musician playing a drum. If you listen closely to the sound it makes, you can often tell something about the drum's shape, the tension of its skin, or even if there is a hidden weight attached to it. In the world of differential equations, the "sound" is a set of frequencies, called eigenvalues, that a system naturally vibrates at. The "drum" is a mathematical model of a physical space, and the "hidden weight" is a potential, a function that describes how energy or matter is distributed within that space. For decades, mathematicians have asked a profound question: if you know the frequencies and how the system behaves at its edges, can you uniquely reconstruct the hidden weight inside? This is the essence of the Borg-Levinson problem, a puzzle that has driven research for nearly a century, moving from simple one-dimensional lines to complex, multi-dimensional shapes.

For a long time, this puzzle was solved only for systems governed by standard, local laws of physics, where the behavior at one point depends only on its immediate neighbors. However, the modern world of physics increasingly deals with "fractional" systems. These are models where the rules are non-local, meaning a point in space can be influenced by distant points, much like how a rumor in a large crowd might skip over several people to reach someone far away. These fractional models are crucial for describing phenomena like anomalous diffusion in porous rocks or the movement of particles in complex biological tissues. The challenge has been to determine if the classic inverse problem—reconstructing the hidden weight from the sound—still holds true when the underlying physics is fractional.

In this work, researchers Saumyajit Das and Tuhin Ghosh tackle the fractional Borg-Levinson problem. They ask whether the internal potential of a fractional system can be uniquely identified if we know its vibrational frequencies and the specific way those vibrations manifest at the boundary of the domain. The authors focus on a specific range of fractional behavior, where the "fractional exponent" lies between one-half and one. This range is technically difficult to handle but covers many physically relevant scenarios. They also impose a condition that the potential being studied is small and non-negative, ensuring the mathematical landscape remains stable enough to analyze.

The researchers begin by establishing a bridge between two different ways of looking at the problem. First, they consider the static, elliptic version of the equation, which describes a system in equilibrium. They prove that if two different potentials produce the exact same set of frequencies and the same boundary data, then the mathematical maps that translate boundary inputs to boundary outputs must be identical. This is a significant step because it reduces the complex spectral data to a more manageable relationship between inputs and outputs at the edge of the domain.

Next, the authors transform this static problem into a dynamic one. They introduce time, turning the equation into a parabolic heat-like equation that describes how the system evolves. By using a mathematical tool called the Laplace transform, they show that the equality of the boundary maps in the static case forces the boundary maps in the time-dependent case to be equal as well. This allows them to treat the problem as a non-local parabolic inverse problem, where the goal is to recover the potential from the evolution of the system over time.

The final and most critical part of their argument relies on the density of solutions. The researchers demonstrate that by varying the input at the boundary, the resulting states of the system inside the domain can fill the space of all possible functions. In simpler terms, they show that the system is sensitive enough to the boundary inputs that the internal state can be made to look like almost any configuration. Because the boundary data for the two different potentials are identical, and because the system can explore all possible internal states, the only way for the equations to hold true is if the two potentials are actually the same.

The paper concludes with a definitive result: within the specific class of small, non-negative potentials and for fractional exponents between one-half and one, the potential is uniquely determined by the boundary spectral data. The authors prove that if two such potentials generate the same frequencies and the same boundary behavior, they must be identical throughout the entire domain. This finding extends the celebrated Borg-Levinson theorem from the familiar world of local physics into the more complex, non-local realm of fractional calculus, confirming that the hidden structure of these systems can indeed be fully revealed by listening to their edge.

Technical Summary: On Fractional Borg-Levinson Problem

Problem Statement
This article investigates the fractional Borg-Levinson problem, an inverse spectral problem concerning the recovery of potentials from boundary spectral data. The study extends the classical multidimensional Borg-Levinson theorem to the setting of fractional Schrödinger operators. Specifically, the authors consider two operators of the form $(−\Delta)^a + q_i$ defined on a smooth, bounded, connected domain $\Omega \subset \mathbb{R}^N$ , where $a \in (1/2, 1)$ is the fractional exponent and $q_i$ are potentials.

The central question is whether the equality of the Dirichlet spectral data for two such operators implies the equality of the potentials. The spectral data consists of the sequence of Dirichlet eigenvalues $\{\lambda_{n, q_i}\}_{n \in \mathbb{N}}$ and the corresponding Dirichlet boundary traces of the normalized eigenfunctions, denoted as $\frac{\phi_{n, q_i}}{d^{a-1}}|_{\partial \Omega}$ , where $d(x)$ is the distance to the boundary. The problem asks: if $\lambda_{n, q_1} = \lambda_{n, q_2}$ and $\frac{\phi_{n, q_1}}{d^{a-1}}|_{\partial \Omega} = \frac{\phi_{n, q_2}}{d^{a-1}}|_{\partial \Omega}$ for all $n$ , does it follow that $q_1 \equiv q_2$ in $\Omega$ ?

Methodology
The authors employ a strategy that bridges spectral data, elliptic boundary value problems, and non-local parabolic inverse problems. The proof proceeds through the following logical steps:

Equality of Elliptic Dirichlet-to-Neumann (DN) Maps:
The authors first establish that the equality of the spectral data (eigenvalues and boundary traces) implies the equality of the Dirichlet-to-Neumann maps for the associated fractional elliptic operators. This is achieved by constructing solutions to the resolvent equation $(−\Delta)^a v + qv = \lambda v$ using spectral expansions and demonstrating that the boundary Neumann data for the difference of solutions vanishes under the spectral assumptions.
Transition to Parabolic Setting:
Using the Laplace transform, the authors convert the equality of the elliptic DN maps into the equality of the parabolic DN maps for the non-local heat equation:
$\partial_t w + (−\Delta)^a w + qw = 0$
This step relies on the analytic properties of the Laplace transform and the well-posedness of the fractional parabolic equation in appropriate $\mu$ -transmission spaces.
Inverse Problem via Solution Space Density:
The core of the uniqueness proof reduces to a non-local parabolic inverse problem: if two potentials generate the same parabolic DN map, are they identical? To resolve this, the authors utilize a density argument for the solution space of the parabolic equation. They demonstrate that the set of solutions at a fixed time $t$ , generated by varying the boundary source data, is dense in $L^2(\Omega)$ . This density result is derived using boundary control techniques and observability estimates.
Uniqueness under Smallness Assumptions:
The final step involves showing that the density of solutions allows the recovery of the potential difference. The authors prove that if the DN maps are equal, then $(q_1 - q_2)w_1 \equiv 0$ . To conclude $q_1 \equiv q_2$ , they impose specific constraints on the potential $q_1$ : it must be non-negative, and its magnitude and gradient must be sufficiently small (bounded by a constant $\theta$ dependent on the Hardy-Sobolev constant and the domain geometry).

Key Contributions and Results

Extension to Fractional Operators: The paper successfully generalizes the multidimensional Borg-Levinson theorem to fractional Schrödinger operators with the exponent restricted to the interval $a \in (1/2, 1)$ .
Uniqueness Theorem: The main result (Theorem 3) states that for a smooth, bounded, connected domain $\Omega$ , if $q_1$ is non-negative and satisfies the smallness condition $|q_1(x)|, |\nabla q_1(x)| \leq \theta$ (where $\theta$ depends on the domain dimension and the Hardy-Sobolev constant), and if the spectral data for $(−\Delta)^a + q_1$ and $(−\Delta)^a + q_2$ coincide, then $q_1 \equiv q_2$ .
Technical Framework: The work integrates the theory of $\mu$ -transmission spaces (introduced by Grubb and Hörmander) to handle the singular boundary behavior of fractional Laplacian eigenfunctions. It rigorously defines the Dirichlet trace $\frac{u}{d^{a-1}}$ and establishes the well-posedness of the associated boundary value problems.
Reduction to Parabolic Inverse Problem: The authors provide a detailed derivation showing how spectral data equality leads to the equality of parabolic DN maps, thereby reducing the spectral inverse problem to a non-local parabolic inverse problem solvable via solution space density.

Significance and Claims
The authors claim that their work extends local results from classical inverse spectral theory (specifically those by Nachman, Sylvester, Uhlmann, and Novikov) to the non-local fractional setting. The paper emphasizes that the restriction $a \in (1/2, 1)$ is necessary for technical reasons, specifically to ensure the continuous inclusion of the relevant Sobolev spaces into $L^2(\partial \Omega)$ , which is crucial for defining the boundary traces and the DN maps.

The significance of the result lies in establishing a uniqueness theorem for a class of non-local inverse problems, which are increasingly relevant in modeling anomalous diffusion and other physical phenomena described by fractional operators. The authors modestly note that their proof relies on a smallness assumption on one of the potentials, a common feature in such inverse problems to ensure the density of the solution space and avoid issues related to non-uniqueness in the general case. The work does not propose new experimental applications but rather provides a theoretical foundation for the solvability of the fractional Borg-Levinson problem under the specified constraints.

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