scholarly journals Two-dimensional Scattering by a Potential without a Classical Analogue

1999 ◽  
Vol 52 (5) ◽  
pp. 859 ◽  
Author(s):  
I. V. Ponomarev
Keyword(s):  
Quantum Chaos ◽  
Matrix Theory ◽  
Delta Function ◽  
Two Dimensional ◽  
Dirac Delta ◽  
S Matrix ◽  
Classical Analogue ◽  
Scattering Potential ◽  
The One ◽  
The Given

A two-dimensional scattering potential represents the quantum extension of a diffractive lattice: a Dirac delta function with a modulated permeability along the y-axis. This model does not have an explicit classical analogue and quantum effects such as tunneling and diffraction play an important role. An analytical solution for the one-harmonic case is found. For the general case of an arbitrary number of harmonics a simple criterion is derived for the range of parameters where quantum chaos is permitted (but does not necessarily occur). The statistical properties of the S-matrix for the given model have been investigated. The deviations from the usual predictions for irregular scattering in the random matrix theory (RMT) framework have been found and are discussed.

S-matrix Theory

2020 ◽  
pp. 622-675
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Matrix Theory ◽  
Point Of View ◽  
Integrable Models ◽  
Geometrical Interpretation ◽  
Coupling Constants ◽  
Two Dimensional ◽  
Recursive Structure ◽  
Mathematical Point ◽  
S Matrix ◽  
Dynamical Principle

Chapter 17 discusses the S-matrix theory of two-dimensional integrable models. From a mathematical point of view, the two-dimensional nature of the systems and their integrability are the crucial features that lead to important simplifications of the formalism and its successful application. This chapter deals with the analytic theory of the S-matrix of the integrable models. A particular emphasis is put on the dynamical principle of bootstrap, which gives rise to a recursive structure of the amplitudes. It also covers several dynamical quantities, such as mass ratios or three-coupling constants, which have an elegant mathematic formulation that is also of easy geometrical interpretation.

scholarly journals Variational approach to the Schrödinger equation with a delta-function potential

2021 ◽  
Author(s):  
Francisco Marcelo Fernandez
Keyword(s):  
Variational Method ◽  
Delta Function ◽  
Basis Set ◽  
One Dimensional ◽  
Dirac Delta ◽  
Introductory Course ◽  
Relevant Role ◽  
Secular Equations ◽  
The One ◽  
Computer Algebra Software

Abstract We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

From Poincare's Divergences to Quantum Mechanics with Broken Time Symmetry

1997 ◽  
Vol 52 (1-2) ◽  
pp. 37-45
Author(s):  
Ilya Prigogine
Keyword(s):  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Perturbation Technique ◽  
Scattering Problem ◽  
Liouville Space ◽  
Time Symmetry ◽  
Classical Chaos ◽  
S Matrix

AbstractWe discuss the spectral property of unstable dynamical systems in both classical and quantum mechanics. An important class of unstable dynamical systems corresponds to the Large Poincare Systems (LPS). Conventional perturbation technique leads then to divergences. We introduce methods for the elimination of Poincare divergences to obtain a solution of the spectral problem analytic in the coupling constant. To do so, we have to enlarge the class of permissible transformations, to include non-unitary transformations as well as to extend the Hilbert space. A simple example refers to the Friedrichs model, which was studied independently by George Sudarshan and his co-workers. However, our main interest is the irreducible representations in the Liouville space. In these representations the central quantity is the density matrix, and the eigenfunctions of the Liouville operator cannot be expressed in terms of the wave functions. We suggest that this situation corresponds to quantum chaos. Indeed, classical chaos does not mean that Newton's equation becomes "wrong" but that trajectories loose their operational meaning. Similarly, whenever we have an irreducible representation in the Liouville space this means that the wave function description looses its operational meaning. Additional statistical features appear. A simple example corresponds to persistent interactions in the scattering problem which cannot be treated in the frame of usual S-matrix theory.

UNIVERSALITY IN TWO–DIMENSIONAL LANDAU LEVEL LOCALIZATION

1993 ◽  
Vol 07 (07) ◽  
pp. 449-457 ◽  
Author(s):  
DONGZI LIU ◽  
S. DAS SARMA
Keyword(s):  
Landau Level ◽  
Critical Exponents ◽  
Strong Field ◽  
Finite Size ◽  
Delta Function ◽  
Scaling Analysis ◽  
Two Dimensional ◽  
Dimensional Electron ◽  
Scattering Potential ◽  
Dimensional Electron Gas

We show, based on a direct numerical calculation of the Lyapunov exponents of the system and a finite-size single parameter scaling analysis, that the strong-field Landau level localization in a disordered two-dimensional electron gas is non-universal for short-range delta function random scatterers in the sense that the critical exponents in the two lowest Landau levels are substantially different. Inclusion of Landau level coupling and/or consideration of finite range of the random scattering potential in the theory restore the universality and make the computed critical exponents approximately equal.

F.VIII R:Ag in Hepatitis

1979 ◽  
Author(s):  
R. Kotitschke ◽  
J. Scharrer
Keyword(s):  
Chronic Hepatitis ◽  
Blood Donors ◽  
Blood Donation ◽  
Normal Population ◽  
Rabbit Antiserum ◽  
Two Dimensional ◽  
Plastic Plate ◽  
Significant Difference ◽  
Acute Hepatitis B ◽  
The One

F.VIII R:Ag was determined by quantitative immunelectrophoresis (I.E.) with a prefabricated system. The prefabricated system consists of a monospecific f.VIII rabbit antiserum in agarose on a plastic plate for the one and two dimensional immunelectrophoresis. The lognormal distribution of the f.VIII R:Ag concentration in the normal population was confirmed (for n=70 the f.VIII R:Ag in % of normal is = 95.4 ± 31.9). Among the normal population there was no significant difference between blood donors (one blood donation in 8 weeks; for n=43 the f.VIII R:Ag in % of normal is = 95.9 ± 34.0) and non blood donors (n=27;f.VIII R:Ag = 94.6 ± 28.4 %). The f.VIII R:Ag concentration in acute hepatitis B ranged from normal to raised values (for n=10, a factor of 1.8 times of normal was found) and was normal again after health recovery (n=10, the factor was 1.0). in chronic hepatitis the f.VIII R:Ag concentration was raised in the majority of the cases (for n=10, the factor was 3.8). Out of 22 carrier sera 20 showed reduced, 2 elevated levels of the f.VIII R:Ag concentration. in 5 sera no f.VIII R:Ag could be demonstrated. The f.VIII R:Ag concentration was normal for n=10, reduced for n=20 and elevated for n=6 in non A-non B hepatitis (n=36). Contrary to results found in the literature no difference in the electrophoretic mobility of the f.VIII R:Ag was found between hepatitis patients sera and normal sera.

THE EU SUGAR MARKET PROFILE AND ITS MAIN DRIVERS

2015 ◽  
Author(s):  
Lubos SMUTKA ◽  
Irena BENEŠOVÁ ◽  
Patrik ROVNÝ ◽  
Renata MATYSIK-PEJAS
Keyword(s):  
Market Failure ◽  
Current System ◽  
Quota System ◽  
Considerable Debate ◽  
Sugar Market ◽  
The Common ◽  
The One ◽  
Production Capacities ◽  
The Given ◽  
The Eu

Sugar is one of the most important elements in human nutrition. The Common Market Organisation for sugar has been a subject of considerable debate since its establishment in 1968. The European agricultural market has been criticized for its heavy regulations and subsidization. The sugar market is one of the most regulated ones; however, this will change radically in 2017 when the current system of production quotas will end. The current EU sugar market changed is structure during the last several decades. The significant number of companies left the market and EU internal sugar market became more concentrated. The aim of this paper is presentation characteristics of sugar market with respect to the supposed market failure – reduction in competition. The analysis also identifies the main drivers and determinants of the EU especially quota sugar market. In relation to paper’s aim the following results are important. The present conditions of the European sugar market have led to market failure when nearly 75 % (10 million tonnes) of the quota is controlled by five multinational companies only. These multinational alliances (especially German and French one) are also taking control over the production capacities of their subsidiaries. In most countries, this causes serious problems as the given quota is controlled by one or two producers only. This is a significant indicator of market imperfection. The quota system cannot overcome the problem of production quotas on the one hand and the demand on the other; furthermore, it also leads to economic inefficiency. The current EU sugar market is under the control of only Sudzucker, Nordzucker, Pfeifer and Langen, Tereos and ABF.

Quasi-Periodic Control Technique for Minimizing Fuel Consumption During Record Vehicle Competition

2013 ◽  
Vol 60 (2) ◽  
pp. 185-197 ◽  
Author(s):  
Paweł Sulikowski ◽  
Ryszard Maronski
Keyword(s):  
Fuel Consumption ◽  
Optimal Control Theory ◽  
Slope Angle ◽  
Control Technique ◽  
Decision Variables ◽  
Aeronautical Engineering ◽  
The One ◽  
Optimal Driving ◽  
The Given ◽  
Engine Power

The problem of the optimal driving technique during the fuel economy competition is reconsidered. The vehicle is regarded as a particle moving on a trace with a variable slope angle. The fuel consumption is minimized as the vehicle covers the given distance in a given time. It is assumed that the run consists of two recurrent phases: acceleration with a full available engine power and coasting down with the engine turned off. The most fuel-efficient technique for shifting gears during acceleration is found. The decision variables are: the vehicle velocities at which the gears should be shifted, on the one hand, and the vehicle velocities when the engine should be turned on and off, on the other hand. For the data of students’ vehicle representing the Faculty of Power and Aeronautical Engineering it has been found that such driving strategy is more effective in comparison with a constant speed strategy with the engine partly throttled, as well as a strategy resulting from optimal control theory when the engine is still active.

Dirac delta regularization

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Finite Result

I present a finite result for the Dirac delta "function."

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

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