J-matrix method of scattering for inverse-square singular potential with supercritical coupling I. Theory

• URL: http://arxiv.org/abs/2012.07493v2
• Date: Fri, 18 Dec 2020 21:41:49 GMT
• Title: J-matrix method of scattering for inverse-square singular potential with supercritical coupling I. Theory
• Authors: Abdulaziz D. Alhaidari, Hocine Bahlouli, Carlos P. Aparicio, and Saeed M. Al-Marzoug
• Abstract summary: The J-matrix method of scattering was developed to handle regular short-range potentials with applications in atomic, nuclear and molecular physics. We extend our study to include the supercritical coupling in which the coupling parameter strength is less than -1/8. In a follow-up study, we intend to apply the method to obtain scattering information for realistic potential models.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The J-matrix method of scattering was developed to handle regular short-range potentials with applications in atomic, nuclear and molecular physics. Its accuracy, stability, and convergence properties compare favorably with other successful scattering methods. It is an algebraic method, which is built on the utilization of orthogonal polynomials that satisfy three-term recursion relations and on the manipulation of tridiagonal matrices. Recently, we extended the method to the treatment of 1/r^2 singular short-range potentials but confined ourselves to the sub-critical coupling regime where the coupling parameter strength of the 1/r^2 singularity is greater than -1/8. In this work, we expand our study to include the supercritical coupling in which the coupling parameter strength is less than -1/8. However, to accomplish that we had to extend the formulation of the method to objects that satisfy five-term recursion relations and matrices that are penta-diagonal. It is remarkable that we could develop the theory without regularization or self-adjoint extension, which are normally needed in the treatment of such highly singular potentials. Nonetheless, we had to pay the price by extending the formulation of the method into this larger representation and by coping with slower than usual convergence. In a follow-up study, we intend to apply the method to obtain scattering information for realistic potential models.

Related papers

• Diagonalization without Diagonalization: A Direct Optimization Approach for Solid-State Density Functional Theory [8.922374095111797]
  We present a novel approach to address the challenges of variable occupation numbers in direct optimization of density functional theory. Our method incorporates physical constraints on both the eigenfunctions and the occupations into the parameterization. It produces the correct Fermi-Dirac distribution of the occupation numbers and yields band structures consistent with those obtained with SCF methods in Quantum Espresso.
  arXiv  Detail & Related papers  (2024-11-06T11:03:40Z)
• A Unified Theory of Stochastic Proximal Point Methods without Smoothness [52.30944052987393]
  Proximal point methods have attracted considerable interest owing to their numerical stability and robustness against imperfect tuning. This paper presents a comprehensive analysis of a broad range of variations of the proximal point method (SPPM)
  arXiv  Detail & Related papers  (2024-05-24T21:09:19Z)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• Reduced density matrix functional theory from an ab initio seniority-zero wave function: Exact and approximate formulations along adiabatic connection paths [0.0]
  We propose an alternative formulation of reduced density-matrix functional theory (RDMFT) The exact natural orbitals and their occupancies are determined self-consistently from an effective seniority-zero calculation. This information is expected to serve as a guide in the future design of higher-seniority density-matrix functional approximations.
  arXiv  Detail & Related papers  (2022-04-01T21:27:25Z)
• Relative Pose from SIFT Features [50.81749304115036]
  We derive a new linear constraint relating the unknown elements of the fundamental matrix and the orientation and scale. The proposed constraint is tested on a number of problems in a synthetic environment and on publicly available real-world datasets on more than 80000 image pairs.
  arXiv  Detail & Related papers  (2022-03-15T14:16:39Z)
• Direct solution of multiple excitations in a matrix product state with block Lanczos [62.997667081978825]
  We introduce the multi-targeted density matrix renormalization group method that acts on a bundled matrix product state, holding many excitations. A large number of excitations can be obtained at a small bond dimension with highly reliable local observables throughout the chain.
  arXiv  Detail & Related papers  (2021-09-16T18:36:36Z)
• Improving Metric Dimensionality Reduction with Distributed Topology [68.8204255655161]
  DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets.
  arXiv  Detail & Related papers  (2021-06-14T17:19:44Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Non-Convex Exact Community Recovery in Stochastic Block Model [31.221745716673546]
  Community detection in graphs that are generated according to symmetric block models (SBMs) has received much attention lately. We show that in the logarithmic sparsity regime of the problem, with high probability the proposed two-stage method can exactly recover the two communities down to the information-theoretic limit in $mathcalO(nlog2n/loglog n)$ time. We also conduct numerical experiments on both synthetic and real data sets to demonstrate the efficacy of our proposed method and complement our theoretical development.
  arXiv  Detail & Related papers  (2020-06-29T07:03:27Z)
• J-matrix method of scattering in one dimension: The relativistic theory [0.0]
  We make a relativistic extension of the one-dimensional J-matrix method of scattering. The relativistic potential matrix is a combination of vector, scalar, and pseudo-scalar components.
  arXiv  Detail & Related papers  (2020-01-14T19:02:15Z)

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.