Connection formulae for spectral functions associated with singular Dirac equations

2004 ◽  
Vol 134 (1) ◽  
pp. 215-223 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Dirac Equation ◽  
Spectral Function ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Dirac Equations ◽  
Special Cases ◽  
Limit Point Case ◽  
Image Position ◽  
Connection Formulae

We consider the Dirac equation given by with initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

Connection formulae for spectral functions associated with singular Sturm–Liouville equations

2000 ◽  
Vol 130 (1) ◽  
pp. 25-34 ◽  
Author(s):  
D. J. Gilbert ◽  
B. J. Harris
Keyword(s):  
Liouville Equation ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Image Position ◽  
Almost All ◽  
Connection Formulae ◽  
Symmetric Derivative

We consider the Sturm–Liouville equation with the initial condition and suppose that Weyl's limit-point case holds at infinity. Let ρα(μ) be the corresponding spectral function and its symmetric derivative. We show that for almost all μ ∈ R, if exists and is positive for some α ∈ [0, π), then (i) exists and is positive for all β ∈ [0, π), and (ii) for all α1, α2 ∈ (0, π) \ {½ π},

Consequences of the connection formulae for Sturm-Liouville spectral functions

2002 ◽  
Vol 132 (2) ◽  
pp. 387-393 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Explicit Formula ◽  
Liouville Equation ◽  
Liouville Problem ◽  
Initial Condition ◽  
Spectral Functions ◽  
Sturm Liouville ◽  
Image Position ◽  
Connection Formulae ◽  
Special Case

For a special case of the Sturm-Liouville equation, −(py′)′ + qy = λwy on [0, ∞) with the initial condition y(0) cos α + p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that, given the spectral derivative for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine . An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with both finite and positive, then the following inequality holds

A Space Correlation and Its Associated Turbulence Spectral Function

1974 ◽  
Vol 52 (3) ◽  
pp. 219-222
Author(s):  
S. N. Samaddar
Keyword(s):  
Spectral Function ◽  
Isotropic Turbulence ◽  
Spectral Functions ◽  
Space Correlation ◽  
Special Cases

A space correlation associated with an isotropic turbulence spectral function is derived. A few special cases of this spectral function are also discussed. One of these special spectral functions was proposed previously for some valid physical grounds.

Limit-point (LP) criteria for real symmetric differential expressions of order 2n

1981 ◽  
Vol 88 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
H. Kurss ◽  
G. Meyer
Keyword(s):  
Limit Point ◽  
The Other ◽  
Special Cases ◽  
Interval Type ◽  
Image Position ◽  
Positive Coefficients

SynopsisAn interval-type LP criterion foris derived in which “positive” coefficients play a prominent role. When pn = 1 and all the other pi are zero this reduces to a result of Ismagilov (1962). Successive specializations are obtained with the growth of the pi constrained by monomials in x. Previous LP criteria of Everitt (1968) and Hinton (1972, 1974) are shown to be special cases.

The form of the spectral functions associated with a class of Sturm–Liouville equations with integrable coefficient

1987 ◽  
Vol 105 (1) ◽  
pp. 215-227 ◽  
Author(s):  
B.J. Harris
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Spectral Function ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Spectral Functions ◽  
Order Linear ◽  
Sturm Liouville ◽  
Linear Differential ◽  
Image Position

SynopsisWe consider the second order, linear differential equationin the case where, roughly, q ∊ L1 [0, ∞). We devise a representation for the spectral function, τ(t), associated with (*) which is valid for t sufficiently large. Our results are the best possible.

scholarly journals Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 477
Author(s):  
Katarzyna Górska ◽  
Andrzej Horzela
Keyword(s):  
Stochastic Processes ◽  
Spectral Function ◽  
Evolution Equations ◽  
Memory Function ◽  
Relaxation Phenomena ◽  
Stieltjes Function ◽  
Infinitely Divisible ◽  
Spectral Functions ◽  
Debye Relaxation ◽  
Relaxation Functions

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M(t), which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M(t) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k(t). Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

scholarly journals The Dirac Electron Consistent with Proper Gravitational and Electromagnetic Field of the Kerr–Newman Solution

Galaxies ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 18
Author(s):  
Alexander Burinskii
Keyword(s):  
Electromagnetic Field ◽  
Quantum Theory ◽  
Gravitational Field ◽  
Dirac Equation ◽  
Higgs Field ◽  
Vacuum State ◽  
Relativistic String ◽  
Dirac Equations ◽  
Dirac Electron ◽  
Relativistic Scattering

The Dirac electron is considered as a particle-like solution consistent with its own Kerr–Newman (KN) gravitational field. In our previous works we considered the regularized by López KN solution as a bag-like soliton model formed from the Higgs field in a supersymmetric vacuum state. This bag takes the shape of a thin superconducting disk coupled with circular string placed along its perimeter. Using the unique features of the Kerr–Schild coordinate system, which linearizes Dirac equation in KN space, we obtain the solution of the Dirac equations consistent with the KN gravitational and electromagnetic field, and show that the corresponding solution takes the form of a massless relativistic string. Obvious parallelism with Heisenberg and Schrödinger pictures of quantum theory explains remarkable features of the electron in its interaction with gravity and in the relativistic scattering processes.

scholarly journals Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ituen B. Okon ◽  
E. Omugbe ◽  
Akaninyene D. Antia ◽  
C. A. Onate ◽  
Louis E. Akpabio ◽  
...  
Keyword(s):  
Dirac Equation ◽  
Thermodynamic Properties ◽  
Energy Equation ◽  
Pseudospin Symmetry ◽  
Spin Symmetry ◽  
Bound State ◽  
Compact Form ◽  
Relativistic Energy ◽  
Quadratic Potential ◽  
Special Cases

AbstractIn this research article, the modified approximation to the centrifugal barrier term is applied to solve an approximate bound state solutions of Dirac equation for spin and pseudospin symmetries with hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential using parametric Nikiforov–Uvarov method. The energy eigen equation and the unnormalised wave function were presented in closed and compact form. The nonrelativistic energy equation was obtain by applying nonrelativistic limit to the relativistic spin energy eigen equation. Numerical bound state energies were obtained for both the spin symmetry, pseudospin symmetry and the non relativistic energy. The screen parameter in the potential affects the solutions of the spin symmetry and non-relativistic energy in the same manner but in a revised form for the pseudospin symmetry energy equation. In order to ascertain the accuracy of the work, the numerical results obtained was compared to research work of existing literature and the results were found to be in excellent agreement to the existing literature. The partition function and other thermodynamic properties were obtained using the compact form of the nonrelativistic energy equation. The proposed potential model reduces to Hulthen and exponential inversely quadratic potential as special cases. All numerical computations were carried out using Maple 10.0 version and Matlab 9.0 version softwares respectively.

scholarly journals Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1373
Author(s):  
Louis H. Kauffman
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Clifford Algebras ◽  
Discrete Systems ◽  
Complex Numbers ◽  
Majorana Fermions ◽  
Matrix Algebras ◽  
Dirac Equations ◽  
Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

scholarly journals TOPOLOGICAL INSULATOR AND THE DIRAC EQUATION

SPIN ◽  
2011 ◽  
Vol 01 (01) ◽  
pp. 33-44 ◽  
Author(s):  
SHUN-QING SHEN ◽  
WEN-YU SHAN ◽  
HAI-ZHOU LU
Keyword(s):  
Dirac Equation ◽  
Topological Insulator ◽  
Bound States ◽  
Topological Insulators ◽  
Mass Term ◽  
Point Of View ◽  
Quadratic Term ◽  
General Description ◽  
Dirac Equations ◽  
Topological Quantum

We present a general description of topological insulators from the point of view of Dirac equations. The Z2 index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic term in momentum is introduced to correct the mass term m or the band gap of the Dirac equation, i.e., m → m − Bp2, the Z2 index is modified as 1 for mB > 0 and 0 for mB < 0. For a fixed B there exists a topological quantum phase transition from a topologically trivial system to a nontrivial system when the sign of mass m changes. A series of solutions near the boundary in the modified Dirac equation is obtained, which is characteristic of topological insulator. From the solutions of the bound states and the Z2 index we establish a relation between the Dirac equation and topological insulators.

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