scholarly journals The quantum-mechanical Coulomb propagator in an L2 function representation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Rolf Gersbacher ◽  
John T. Broad
Keyword(s):  
Recursion Relation ◽  
Contour Integral ◽  
Quantum Mechanical ◽  
Stieltjes Integral ◽  
Pollaczek Polynomials ◽  
Discrete Part ◽  
Legendre Quadrature ◽  
The Continuum ◽  
Square Integrable Basis ◽  
Sturmian Functions

AbstractThe quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L$$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

A micromechanically calibrated numerical model reproducing earthquake cycle in the lab

2021 ◽  
Author(s):  
Guilhem Mollon ◽  
Jérôme Aubry ◽  
Alexandre Schubnel
Keyword(s):  
Acoustic Waves ◽  
Granular Matter ◽  
Fluid Pressure ◽  
Triaxial Tests ◽  
Zone Model ◽  
Geophysical Research ◽  
Stick Slip ◽  
Space And Time ◽  
Discrete Part ◽  
The Continuum

<p>In this communication, we present a novel numerical framework which consists in a direct coupling between a discrete micromechanical modelling of rock damaging processes and a continuous modelling of elastic deformation and acoustic waves. It includes a polygon-based conforming Discrete Element Method (DEM) with a cohesive zone model (CZM, [1]) for the discrete part and a meshfree formulation for the continuum part. This framework is applied to the numerical reproduction of sawcut triaxial tests performed in the lab on marble samples under seismogenic conditions [2]. Realistic boundary conditions (in terms of the elasticity of the loading system, of the absorption of the elastic waves and of the fluid pressure applied on the lateral boundaries) are introduced. Constitutive laws (in the continuum part) and micromechanical parameters (in the discrete part) are calibrated by performing independant simulations based on experimental results found in the literature [3].</p><p>Upon loading, this model provides information on the system behavior that nicely complement the experimental data, such as (i) the progressive damaging of the contacting surfaces, leading to the emission of granular matter in the interface, to the formation of a gouge layer, and to a modification of the interface rheology, (ii) the space and time distribution and statistics and the detailed kinematics of the slip events related to the interface evolution, and (iii) the acoustic wave emission and propagation in the medium associated with such events.</p><p>The model shows that, depending on the experimental conditions (confining pressure, loading rate, surface roughness, etc.), and without relying to any prior choice of slip- or rate-dependent friction laws, a large number of sliding regimes can emerge from this system. This includes large stress drops, regular stick-slip, or stable sliding. This model thus provides an unprecedented view of both local and global phenomena at stake during lab earthquakes, at sampling rates in both space and time which remain out of reach for experimental instrumentation.</p><p>[1]. Mollon, G. (2015). “A numerical framework for discrete modelling of friction and wear using Voronoi polyhedrons”, Tribology International, 90, 343-355<br>[2]. Aubry, J. (2019). “Séismes au laboratoire: friction, plasticité et bilan énergétique”, PhD Thesis, Ecole Normale Supérieure.<br>[3]. Fredrich, J. T.; Evans, B. & Wong, T.-F., (1989). “Micromechanics of the brittle to plastic transition in Carrara marble”, Journal of Geophysical Research: Solid Earth, <span></span></p>

THE (ORBIFOLD) EULER CHARACTERISTIC OF THE MODULI SPACE OF CURVES AND THE CONTINUUM LIMIT OF PENNER'S CONNECTED GENERATING FUNCTION

1991 ◽  
Vol 03 (03) ◽  
pp. 285-300 ◽  
Author(s):  
NOUREDDINE CHAIR
Keyword(s):  
Free Energy ◽  
Matrix Models ◽  
Euler Characteristic ◽  
Generating Function ◽  
Moduli Space ◽  
Continuum Limit ◽  
Recursion Relation ◽  
Moduli Space Of Curves ◽  
Genus Zero ◽  
The Continuum

The generating function that gives rise to the orbifold Euler characteristic of the moduli space of punctured compact Rieman surfaces [Formula: see text], g ≥ 0 is derived explicitly. In the derivation, we show that we do not need to use the three-term recursion relation for the orthogonal polynomials. Also the continuum limit of Penner's connected generating function is considered and is shown to be formally equivalent to the free energy obtained recently by Distler and Vafa which exhibits the logarithmic divergences found for genus zero and one in D = 1 matrix models. Finally, it is shown that the free energy and its s-derivatives are nothing but the continuum limit of a certain generating function introduced by Harer and Zagier in obtaining the true Euler characteristic with any number of punctures,[Formula: see text], s ≥ 0.

Diffraction of axisymmetric waves in a borehole by bed boundary discontinuities

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1586-1595 ◽  
Author(s):  
W. C. Chew ◽  
S. Barone ◽  
B. Anderson ◽  
C. Hennessy
Keyword(s):  
Scattering Problem ◽  
Axially Symmetric ◽  
The Finite Element Method ◽  
Mode Spectrum ◽  
The Past ◽  
Symmetric Source ◽  
Discrete Part ◽  
Computational Resources ◽  
Borehole Waves ◽  
The Continuum

This paper presents the calculation of the diffraction of axisymmetric borehole waves by bed boundary discontinuities. The bed boundary is assumed to be horizontal and the inhomogeneities to be axially symmetric. In such a geometry, an axially symmetric source will produce only axially symmetric waves. Since the borehole is an open structure, the mode spectrum consists of a discrete part as well as a continuum. The scattering of a continuum of waves by bed boundaries is difficult to treat. The approach used in the past in treating this class of problem has been approximate in nature, or highly numerical, such as the finite‐element method. We present here a systematic way to approximate the continuum of modes by discrete modes. After discretization, the scattering problem can be treated simply. Since the approach is systematic, it allows derivation of the solution to any desired degree of accuracy in theory; but in practice, it is limited by the computational resources available. We also show that our approach is variational and satisfies both the reciprocity theorem and energy conservation.

scholarly journals Continuum models and Morse Theory in the simulation, design and evaluation of magnetic Skyrmion devices

2021 ◽  
Author(s):  
P. Robert Kotiuga
Keyword(s):  
Morse Theory ◽  
Lattice Models ◽  
Energy Functional ◽  
Continuum Models ◽  
Quantum Mechanical ◽  
Simulation Design ◽  
Lattice Systems ◽  
Solid Foundation ◽  
Magnetic Skyrmions ◽  
The Continuum

Nanomagnetic devices such as computer gates and memory devices based on magnetic skyrmions are close to becoming a reality. In this paper we will explore the highly nonconvex nanomagnetic energy landscape in order to draw conclusions about the complexity of magnetic phenomena. Morse theoretic arguments show that, in a bounded energy interval, the number of critical points of the energy functional grows exponentially. To show this, we introduce a hierarchy of models for the design of nanomagnetic devices to provide a solid foundation for the introduction of topological tools. To reason in terms of lattice models, one must make a distinction between two types of lattices: the quantum mechanical model of the actual physical lattice and the lattice model that can be associated with a discretization of a continuum model of the physics. By focusing on the implications of Morse theory applied to lattice systems arising from the discretization of the continuum models, and the notion of “topological frustration”, we provide a framework for understanding “complexity” in the context of nanomagnetic systems. We conclude with some suggestions for making the analysis more qualitative.

scholarly journals Continuum models and Morse Theory in the simulation, design and evaluation of magnetic Skyrmion devices

2021 ◽  
Author(s):  
P. Robert Kotiuga
Keyword(s):  
Morse Theory ◽  
Lattice Models ◽  
Energy Functional ◽  
Continuum Models ◽  
Quantum Mechanical ◽  
Simulation Design ◽  
Lattice Systems ◽  
Solid Foundation ◽  
Magnetic Skyrmions ◽  
The Continuum

Nanomagnetic devices such as computer gates and memory devices based on magnetic skyrmions are close to becoming a reality. In this paper we will explore the highly nonconvex nanomagnetic energy landscape in order to draw conclusions about the complexity of magnetic phenomena. Morse theoretic arguments show that, in a bounded energy interval, the number of critical points of the energy functional grows exponentially. To show this, we introduce a hierarchy of models for the design of nanomagnetic devices to provide a solid foundation for the introduction of topological tools. To reason in terms of lattice models, one must make a distinction between two types of lattices: the quantum mechanical model of the actual physical lattice and the lattice model that can be associated with a discretization of a continuum model of the physics. By focusing on the implications of Morse theory applied to lattice systems arising from the discretization of the continuum models, and the notion of “topological frustration”, we provide a framework for understanding “complexity” in the context of nanomagnetic systems. We conclude with some suggestions for making the analysis more qualitative.

Bound states in the continuum in asymmetrical quantum-mechanical and electromagnetic waveguides

2021 ◽  
Vol 104 (12) ◽  
Author(s):  
N. M. Shubin ◽  
A. V. Friman ◽  
V. V. Kapaev ◽  
A. A. Gorbatsevich
Keyword(s):  
Bound States ◽  
Quantum Mechanical ◽  
The Continuum

scholarly journals THE BERRY PHASE AND MONOPOLES IN NON-ABELIAN GAUGE THEORIES

2002 ◽  
Vol 17 (02) ◽  
pp. 157-174 ◽  
Author(s):  
F. V. GUBAREV ◽  
V. I. ZAKHAROV
Keyword(s):  
Gauge Theory ◽  
Wilson Loop ◽  
Continuum Limit ◽  
Gauge Theories ◽  
Berry Phase ◽  
Quantum Mechanical ◽  
Abelian Gauge ◽  
Dynamical Phase ◽  
Physical Consequences ◽  
The Continuum

We consider the quantum mechanical notion of the geometrical (Berry) phase in SU(2) gauge theory, both in the continuum and on the lattice. It is shown that in the coherent state basis eigenvalues of the Wilson loop operator naturally decompose into the geometrical and dynamical phase factors. Moreover, for each Wilson loop there is a unique choice of U(1) gauge rotations which do not change the value of the Berry phase. Determining this U(1) locally in terms of infinitesimal Wilson loops we define monopole-like defects and study their properties in numerical simulations on the lattice. The construction is gauge dependent, as is common for all known definitions of monopoles. We argue that for physical applications the use of the Lorentz gauge is most appropriate. And, indeed, the constructed monopoles have the correct continuum limit in this gauge. Physical consequences are briefly discussed.

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