Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities

• URL: http://arxiv.org/abs/2606.01718v1
• Date: Mon, 01 Jun 2026 05:33:58 GMT
• Title: Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities
• Abstract summary: This paper studies the Fourier spectral method (FSM) for the Schrdinger equation with singular potentials $V in Hs$.<n>In the presence of potentials with isolated point singularities, we develop an technique to post-process the FSM solutions.<n>The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders $2s+2+2b$ for eigenvalues and $s+1+b$ for eigenfunctions in the $H1$ norm.
• Score: 10.172011154321908
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials $V \in H^{s}$, where $s > \max\{d/2-2,-1\}$ and $d$ denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order $2s+2$ for eigenvalues and order $s+1$ for eigenfunctions in the $H^1$ norm. More importantly, the $H^1$ error with respect to the projected eigenfunction converges with a higher order $s+1+b$, where $b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0$ for arbitrarily small $\varepsilon>0$, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders $2s+2+2b$ for eigenvalues and $s+1+b$ for eigenfunctions in the $H^1$ norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in $H^{s+4}$ together with $d+1$ asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

Related papers

• Scale-Invariant Open Quantum Systems [0.0]
  We develop a complete theoretical framework for open quantum systems coupled to scale-invariant environments.<n>This work provides the proof of the uniqueness theorem, the formalism of the resulting non-Markovian dynamics.<n>We discuss experimental implications for trapped-ion simulators, neutrino telescopes, and superconducting qubits.
  arXiv  Detail & Related papers  (2026-05-21T18:00:36Z)
• Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation [0.0]
  We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE.<n>Across a sweep of model sizes, we obtain test $H1$-errors down to $mathcalO(10-7)$ and relative errors of order $10-3$, with predictions accurately matching both solutions and spatial derivatives.
  arXiv  Detail & Related papers  (2026-05-04T22:15:21Z)
• Discrete symmetries in classical and quantum oscillators [51.56484100374058]
  We show the eigenfunctions $_n=zn$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation.<n>The superposition $=sum_n c_n_n$ arises only with incomplete knowledge of the initial data for solving the Schrdinger equation.
  arXiv  Detail & Related papers  (2026-01-05T10:04:39Z)
• Field digitization scaling in a $\mathbb{Z}_N \subset U(1)$ symmetric model [0.0]
  We propose to analyze field digitization by interpreting the parameter $N$ as a coupling in the renormalization group sense.<n>Using effective field theory, we derive generalized scaling hypotheses involving the FD parameter $N$.<n>We analytically prove that our calculations for the 2D classical-statistical $mathbbZ_N$ clock model are directly related to the quantum physics in the ground state of a (2+1)D $mathbbZ_N$ lattice gauge theory.
  arXiv  Detail & Related papers  (2025-07-30T18:00:02Z)
• Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel [0.0]
  We analyze the phase-space structure of Schr"odinger $U(D)$-spin cat states.<n>We compute the generalized heat kernel relating two quasi-probability distributions $mathcalF(s)_rho$ and $mathcalF(s')_rho$.
  arXiv  Detail & Related papers  (2025-07-20T08:26:28Z)
• On the second-order zero differential properties of several classes of power functions over finite fields [4.100056500795057]
  Feistel Boomerang Connectivity Table (FBCT) is an important cryptanalytic technique on analysing the resistance of the Feistel network-based ciphers to power attacks such as differential and boomerang attacks. In this paper, by computing the number of solutions of specific equations over finite fields, we determine explicitly the second-order zero differential spectra of power functions $x2m+3$ and $x2m+5$. The computation of these entries and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes.
  arXiv  Detail & Related papers  (2024-09-18T04:27:03Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Stochastic behavior of outcome of Schur-Weyl duality measurement [45.41082277680607]
  We focus on the measurement defined by the decomposition based on Schur-Weyl duality on $n$ qubits. We derive various types of distribution including a kind of central limit when $n$ goes to infinity.
  arXiv  Detail & Related papers  (2021-04-26T15:03:08Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)
• Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions [84.49087114959872]
  We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonsmooth functions. In particular, we study Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions.
  arXiv  Detail & Related papers  (2020-02-10T23:23:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.