https://doi.org/10.1140/epjc/s10052-025-15130-9
Regular Article -Theoretical Physics
Ill-posedness in limited discrete Fourier inversion and regularization for quasi distributions in LaMET
1
Frontiers Science Center for Rare Isotopes, and School of Nuclear Science and Technology, Lanzhou University, 730000, Lanzhou, China
2
Key Laboratory of Atomic and Subatomic Structure and Quantum Control (MOE), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Institute of Quantum Matter, 510006, Guangzhou, China
3
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Guangdong Provincial Key Laboratory of Nuclear Science, Southern Nuclear Science Computing Center, South China Normal University, 510006, Guangzhou, China
4
School of Mathematics and Statistics, Lanzhou University, 730000, Lanzhou, China
5
School of Physics, Beihang University, 102206, Beijing, China
a
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b
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Received:
14
August
2025
Accepted:
28
November
2025
Published online:
11
December
2025
Abstract
We systematically investigated the limited inverse discrete Fourier transform of the quasi distributions from the perspective of inverse problem theory. This transformation satisfies two of Hadamard’s well-posedness criteria, existence and uniqueness of solutions, but critically violates the stability requirement, exhibiting exponential sensitivity to input perturbations. To address this instability, we implemented Tikhonov regularization with L-curve optimized parameters, demonstrating its validity for controlled toy model studies and real lattice QCD results of quasi distribution amplitudes. The reconstructed solutions is consistent with the physics-driven 
-extrapolation method. Our analysis demonstrates that the inverse Fourier problem within the large-momentum effective theory (LaMET) framework belongs to a class of moderately tractable ill-posed problems, characterized by distinct spectral properties that differ from those of more severely unstable inverse problems encountered in other lattice QCD applications. Tikhonov regularization establishes a rigorous mathematical framework for addressing the underlying instability, enabling first-principles uncertainty quantification without relying on ansatz-based assumptions.
© The Author(s) 2025
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Funded by SCOAP3.
