Unlocking Potential: Recovering Quantum Systems from Transition Data

Author: Denis Avetisyan

New research demonstrates a powerful method for uniquely determining a quantum potential by analyzing how a system evolves between initial and final states.

The study delineates the bounds of applicability for critical function space estimations-specifically, for <span class="katex-eq" data-katex-display="false">L^{q}(\mathbb{R}^{n})</span> potentials, Strichartz pairs <span class="katex-eq" data-katex-display="false">(r,p)</span>, and the Stein-Tomas extension <span class="katex-eq" data-katex-display="false">L^{2}(\mathbb{S}^{n-1})\to L^{p}(\mathbb{R}^{n})</span>-demonstrating that the Kenig-Ruiz-Sogge estimate holds for <span class="katex-eq" data-katex-display="false">p\in[q\_{2},\in fty)</span> in two dimensions and <span class="katex-eq" data-katex-display="false">p\in[q\_{n},p\_{n}]</span> for higher dimensions, thereby establishing a nuanced understanding of their dimensional dependence. — The study delineates the bounds of applicability for critical function space estimations-specifically, for $L^{q}(\mathbb{R}^{n})$ potentials, Strichartz pairs $(r,p)$ , and the Stein-Tomas extension $L^{2}(\mathbb{S}^{n-1})\to L^{p}(\mathbb{R}^{n})$ -demonstrating that the Kenig-Ruiz-Sogge estimate holds for $p\in[q\_{2},\in fty)$ in two dimensions and $p\in[q\_{n},p\_{n}]$ for higher dimensions, thereby establishing a nuanced understanding of their dimensional dependence.

This work proves unique recoverability of time-independent potentials from initial-to-final maps, even with weak decay assumptions, through a novel orthogonality relation and refined spectral analysis.

Determining a potential from its effect on quantum states-the inverse problem-remains a significant challenge, particularly when potentials exhibit singularities. This paper, ‘The initial-to-final-state inverse problem with critically-singular potentials’, addresses this issue by establishing conditions under which the time-independent potential $V$ in the Schrödinger equation is uniquely determined by knowledge of the initial-to-final state map. Specifically, uniqueness is proven for potentials satisfying $V \in L^1(\mathbb{R}^n) \cap L^q(\mathbb{R}^n)$ under appropriate conditions on $q$ , extending prior work with weaker decay assumptions through a refined resolvent estimate. Could these results pave the way for recovering more general potentials from limited quantum evolution data?

The Inverse Problem: Echoes of Systemic Decay

The task of discerning the potential energy landscape that dictates a quantum system’s behavior represents a core challenge known as an inverse problem, resonating far beyond theoretical physics. Unlike directly solving the Schrödinger equation with a known potential, this problem requires inferring the potential itself from observed quantum phenomena – a significantly more complex undertaking. This has profound implications for diverse fields; in medical imaging, reconstructing the internal structure of tissues relies on solving an inverse problem using wave propagation, analogous to quantum mechanics. Similarly, seismology uses wave patterns to map the Earth’s interior, and in materials science, determining a material’s atomic potential is crucial for predicting its properties. Effectively solving this inverse problem unlocks the ability to not only understand existing quantum systems but also to design and engineer novel materials and technologies with tailored characteristics, making it a central pursuit in modern scientific inquiry.

Conventional techniques for determining the potential energy governing a quantum system frequently encounter challenges stemming from the inherent ‘ill-posedness’ of the problem – meaning a small error in observed data can lead to wildly different potential energy landscapes. This sensitivity necessitates imposing strong, often unrealistic, assumptions about the potential’s smoothness – its ‘regularity’ – to arrive at a stable solution. For instance, physicists might assume the potential is continuously differentiable or even analytic, effectively pre-defining the characteristics of the system being investigated. While simplifying the mathematics, such constraints risk masking the true, potentially complex, behavior of the quantum system and limiting the accuracy of the recovered potential, particularly in scenarios where the actual potential contains discontinuities or sharp features. This reliance on strong assumptions highlights a fundamental limitation of traditional approaches and motivates the development of more robust and data-driven methods.

The challenge of reconstructing a quantum potential from limited observational data necessitates a departure from conventional analytical techniques. Existing methods frequently falter when faced with incomplete datasets, highlighting the need for innovative mathematical frameworks. Researchers are increasingly exploring techniques like regularization methods and sparse recovery algorithms, alongside advancements in integral equation theory, to stabilize the recovery process. These approaches aim to impose physically plausible constraints on the potential, effectively navigating the inherent ill-posedness of the inverse problem. Furthermore, the development of robust analytical tools-including sophisticated numerical simulations and advanced optimization algorithms-is crucial for accurately estimating the potential and validating the reliability of recovered solutions, paving the way for advancements in quantum technology and fundamental physics.

Mathematical Foundations: The Language of Quantum States

Solutions to the Schrödinger equation are rigorously defined and analyzed within the framework of Lebesgue spaces, particularly $L^1$ . The requirement that the potential, $V$ , belongs to $L^1$ -meaning the integral of its absolute value over all space is finite-is critical for establishing the existence and uniqueness of solutions. This integrability condition ensures that the potential’s contribution to the Hamiltonian operator is well-defined and prevents divergences in the calculations. Furthermore, belonging to $L^1$ facilitates the application of various mathematical tools, such as the spectral theorem and functional analysis techniques, which are essential for studying the equation’s properties and obtaining meaningful physical results. Without this integrability condition, the Schrödinger equation may not have a well-defined solution, or the solution may not possess the necessary properties for physical interpretation.

The analysis of stationary states in quantum mechanics, derived from solutions to the time-independent Schrödinger equation $iħ\partial_t \psi = H\psi$ , is significantly streamlined by recognizing the equivalence to the Helmholtz equation. This transformation allows for the application of established mathematical techniques developed for wave propagation problems. Specifically, solving the time-independent equation $H\psi = E\psi$ yields solutions of the form $\psi(x) = e^{-iEt/ħ} \phi(x)$ , where $\phi(x)$ satisfies the Helmholtz equation $(-\Delta + V(x)) \phi(x) = E \phi(x)$ . This reduction simplifies the mathematical treatment, enabling the use of spectral analysis and functional analysis tools to understand the energy eigenvalues and corresponding stationary state wavefunctions.

The analysis of potential decay rates relies on establishing the potential, $V$ , as a member of specific Lebesgue spaces. Specifically, it is demonstrated that for $n \geq 3$ dimensions, the potential must belong to both $L^1(\mathbb{R}^n)$ and $L^q(\mathbb{R}^n)$ , where $q > n/2$ . For the two-dimensional case, $n = 2$ , the potential is required to be in both $L^1(\mathbb{R}^2)$ and $L^q(\mathbb{R}^2)$ , with the condition that $q > 1$ . These conditions on $q$ are critical for ensuring the well-posedness of the Schrödinger equation and for establishing bounds on the solutions, particularly in endpoint cases where traditional methods may fail.

Strichartz pairs, consisting of function spaces $X$ and $Y$ satisfying the homogeneity condition $\frac{n}{2} < s \le \frac{n}{2} + 1$ , provide a framework for analyzing the time-dependent Schrödinger equation $i\partial_t u = - \Delta u + V(x)u$ . These pairs are characterized by their ability to provide both local and global well-posedness for the equation under certain conditions on the potential $V$ . Specifically, solutions $u(t,x)$ are often estimated within these spaces using Strichartz estimates, which bound the $L^p$ norm of the solution in time and the $L^q$ norm in space. The selection of appropriate Strichartz pairs is crucial for establishing the existence, uniqueness, and regularity of solutions, thereby bolstering the reliability and scope of the analytical methods employed.

Reconstructing the Potential: An Orthogonality-Based Approach

The Alessandrini-type orthogonality relation establishes a direct connection between the initial-to-final state map, denoted as $F$ , and the potential $V$ responsible for scattering. Specifically, this relation arises from the property that the difference between the wave functions generated by two different potentials must be orthogonal to incoming waves. Mathematically, this orthogonality is expressed as an integral condition involving the difference in the potentials and the incoming wave function. This connection is pivotal because it transforms the problem of uniquely determining the potential $V$ from scattering data – represented by the initial-to-final state map $F$ – into a mathematical equation relating the two. By analyzing this integral relation, it becomes possible to investigate the conditions under which the potential can be uniquely recovered from the observed scattering behavior.

Applying the Alessandrini-type orthogonality relation to stationary states-solutions to the time-independent Schrödinger equation-yields a specific integral equation that directly links the potential $V$ to the initial-to-final state map. This formulation arises from the orthogonality condition requiring that the wavefunction associated with a given energy $E$ is orthogonal to any wavefunction of a different energy. Mathematically, this translates to an integral equation of the form $\intV(x)ψ(x)dx = 0$ for all ψ representing states orthogonal to the reference state, where the integral is taken over the spatial domain. Solving this integral equation-though often challenging-provides a pathway to determine the potential $V$ given knowledge of the scattering data encoded in the initial-to-final state map.

The Kenig-Ruiz-Sogge estimate, applicable to solutions of the Helmholtz equation $(Δ + λ^2)u = 0$ , establishes bounds on the $L^p$ norms of solutions in terms of the $L^p$ norms of the forcing term. Specifically, it provides estimates of the form $||u||_{L^p} \lesssim ||f||_{L^p}$ under certain conditions on $p$ and the smoothness of the boundary. These bounds are crucial for demonstrating the well-posedness of the integral equation derived for the potential, as they ensure that solutions exist, are unique, and depend continuously on the input data, specifically the initial-to-final state map. Without these established bounds, proving the existence and uniqueness of the potential recovery becomes significantly more challenging.

The uniqueness of the potential recovery is established through an analysis of the difference between two potentials, $V_1$ and $V_2$ , that produce identical initial-to-final state maps. It is demonstrated that this difference vanishes as the energy parameter λ approaches infinity, with the rate of convergence quantified by the bound $|F̂(ξ)| ≲ λ^{-2n+1}(‖V_1‖_{L^1(ℝ^n)∩L^{n/2}(ℝ^n)} + ‖V_2‖_{L^1(ℝ^n)∩L^{n/2}(ℝ^n)})^2(1 + ‖F‖_λ)$ . This inequality indicates that the magnitude of the Fourier transform of the difference between the potentials, $F̂(ξ)$ , is bounded above by a function of λ, the L¹ and L^n/2 norms of both potentials, and the norm of the initial-to-final state map $F$ . The exponent $-2n+1$ signifies the decay rate with increasing λ, providing a precise measure of the potential’s uniqueness.

The pursuit of uniquely determining a potential from limited initial-to-final state information, as demonstrated in this work, echoes a fundamental principle of all systems: their inherent impermanence. The paper’s rigorous establishment of a novel orthogonality relation, allowing for the reconstruction of time-independent potentials, feels less like achieving stability and more like precisely mapping a system’s decay. Pierre Curie observed, “Nothing can be said to be certain, but everything may be said to be probable.” This aligns with the article’s core idea – that even with relaxed decay assumptions on the potential, a unique solution can be probabilistically determined through spectral analysis. The work doesn’t halt decay, but it meticulously charts its course, revealing the underlying structure even as the system evolves.

What Lies Ahead?

The demonstration that a potential gracefully relinquishes its secrets via the initial-to-final map is, predictably, not a final accounting. This work establishes a recovery process, yet the chronicle of that process – the logging of iterative refinement – remains largely unwritten. Future efforts will inevitably turn to the quality of that recovery; how quickly, and with what fidelity, can the potential be reconstructed from imperfect observations? The current analysis hinges on detailed spectral properties, and the sensitivity of this reconstruction to deviations from ideal conditions warrants investigation.

Furthermore, this paper addresses the time-independent potential. The deployment of these techniques to time-dependent scenarios represents a natural, though daunting, progression. The orthogonality relation, established here as a static property, may hold echoes in time-dependent regimes, but its adaptation will require a reassessment of the fundamental assumptions. The timeline of potential recovery will undoubtedly become more complex, yet the underlying principle – that the system’s chronicle contains its definition – should remain constant.

Ultimately, the question isn’t merely can a potential be determined, but how robust is that determination against the inevitable decay of information. Every measurement introduces a perturbation, every reconstruction an approximation. The true challenge lies not in solving the inverse problem, but in understanding the limits of solvability itself.

Original article: https://arxiv.org/pdf/2602.12122.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Inverse Problem: Echoes of Systemic Decay

Mathematical Foundations: The Language of Quantum States

Reconstructing the Potential: An Orthogonality-Based Approach

What Lies Ahead?

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