scholarly journals Some Applications of the Spectral Theory for the Integral Transform Involving the Spectral Representation

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hyun Soo Chung ◽  
Seung Jun Chang
Keyword(s):  
Quantum Mechanics ◽  
Function Space ◽  
Spectral Theory ◽  
Spectral Representation ◽  
Integral Transform ◽  
Adjoint Operator ◽  
Function Spaces

In many previous papers, an integral transformℱγ,βwas just considered as a transform on appropriate function spaces. In this paper we deal with the integral transform as an operator on a function space. We then apply various operator theories toℱγ,β. Finally we give an application for the spectral representation of a self-adjoint operator which plays a key role in quantum mechanics.

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

scholarly journals Duals of vector-valued Köthe function spaces

1992 ◽  
Vol 112 (1) ◽  
pp. 165-174 ◽  
Author(s):  
Miguel Florencio ◽  
Pedro J. Paúl ◽  
Carmen Sáez
Keyword(s):  
Function Space ◽  
Normed Space ◽  
Function Spaces ◽  
Integrable Functions ◽  
Nikodym Property ◽  
Köthe Dual ◽  
Vector Valued

AbstractLet Λ be a perfect Köthe function space in the sense of Dieudonné, and Λ× its Köthe-dual. Let E be a normed space. Then the topological dual of the space Λ(E) of Λ-Bochner integrable functions equals the corresponding Λ×(E′) if and only if E′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind.

Spectral generating operators for non-stationary processes

1976 ◽  
Vol 8 (4) ◽  
pp. 831-846 ◽  
Author(s):  
D. Tj⊘stheim
Keyword(s):  
Spectral Representation ◽  
Stationary Process ◽  
Adjoint Operator ◽  
Stationary Processes ◽  
Wide Sense ◽  
Linear Transformations ◽  
Generating Operators ◽  
The Time Domain ◽  
Image Position ◽  
Stationary Random Processes

A new method for obtaining spectral-like representations for a large class of non-stationary random processes is formulated. For a wide sense stationary process X(t) in continuous-time the spectral representation is generated by a self-adjoint operator H such that X(t)= eiHt X(0). Extending certain recently established operator identities for wide sense stationary processes, it is shown that similar operators exist for classes of non-stationary processes. The representation generated by such an operator has the form and it shares some of the properties of the wide sense stationary spectral representation: it is dual in a precisely defined sense to the time domain representation of X(t). There exists a class L of linear transformations of X(t) such that for G ∈ L for some function g determined by G.

Separation axioms on function spaces defined on bitopological spaces

2018 ◽  
Vol 9 (2) ◽  
pp. 113-118
Author(s):  
N. E. Muturi ◽  
J. M. Khalagai ◽  
G. P. Pokhariyal
Keyword(s):  
Function Space ◽  
Function Spaces ◽  
Separation Axioms ◽  
Bitopological Spaces

In this paper, we introduce separation axioms on the function space p− Cω(Y, Z) and study how they relateto separation axioms defined on the spaces (Z, δi) for i = 1, 2, (Z, δ1, δ2), 1 − Cς(Y, Z) and 2 − Cζ(Y, Z). Itis shown that the space p − Cω(Y, Z) is pT◦, pT1, pT2 and pregular, if the spaces (Z, δ1) and (Z, δ2) are bothT0, T1, T2 and regular respectively. The space p − Cω(Y, Z) is also shown to be pT0, pT1, pT2 and pregular,if the space (Z, δ1, δ2) is p − T0, p − T1, p − T2 and p-regular respectively. Finally, the space p − Cω(Y, Z) isshown to be pT0, pT1, pT2 and pregular, if and only if the spaces 1 − Cς(Y, Z) and 2 − Cζ(Y, Z) are both T0,T1, T2, and only if the spaces 1 − Cς(Y, Z) and 2 − Cζ(Y, Z) are both regular respectively.

scholarly journals On the spectral theory of a tapered rod

2002 ◽  
Vol 132 (3) ◽  
pp. 729-764 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Function Space ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Boundary Value ◽  
Point Of View ◽  
Point Boundary ◽  
Cross Sectional ◽  
Mechanical Interpretation ◽  
The One ◽  
Image Position

This paper establishes the main features of the spectral theory for the singular two-point boundary-value problem and which models the buckling of a rod whose cross-sectional area decays to zero at one end. The degree of tapering is related to the rate at which the coefficient A tends to zero as s approaches 0. We say that there is tapering of order p ≥ 0 when A ∈ C([0, 1]) with A(s) > 0 for s ∈ (0, 1] and there is a constant L ∈ (0, ∞) such that lims→0A(s)/sp = L. A rigorous spectral theory involves relating (1)−(3) to the spectrum of a linear operator in a function space and then investigating the spectrum of that operator. We do this in two different (but, as we show, equivalent) settings, each of which is natural from a certain point of view. The main conclusion is that the spectral properties of the problem for tapering of order p = 2 are very different from what occurs for p < 2. For p = 2, there is a non-trivial essential spectrum and possibly no eigenvalues, whereas for p < 2, the whole spectrum consists of a sequence of simple eigenvalues. Establishing the details of this spectral theory is an important step in the study of the corresponding nonlinear model. The first function space that we choose is the one best suited to the mechanical interpretation of the problem and the one that is used for treating the nonlinear problem. However, we relate this formulation in a precise way to the usual L2 setting that is most common when dealing with boundary-value problems.

QUANTUM DEFORMATIONS OF QUANTUM MECHANICS

1993 ◽  
Vol 08 (01) ◽  
pp. 89-96 ◽  
Author(s):  
MARCELO R. UBRIACO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Time Evolution ◽  
Function Space ◽  
Scalar Product ◽  
Hypergeometric Functions ◽  
Quantum Mechanical ◽  
Basic Hypergeometric Functions ◽  
Quantum Deformations ◽  
The One

Based on a deformation of the quantum mechanical phase space we study q-deformations of quantum mechanics for qk=1 and 0<q<1. After defining a q-analog of the scalar product on the function space we discuss and compare the time evolution of operators in both cases. A formulation of quantum mechanics for qk=1 is given and the dynamics for the free Hamiltonian is studied. For 0<q<1 we develop a deformation of quantum mechanics and the cases of the free Hamiltonian and the one with a x2-potential are solved in terms of basic hypergeometric functions.

scholarly journals A NECESSARY AND SUFFICIENT CONDITION FOR CERTAIN MARTINGALE INEQUALITIES IN BANACH FUNCTION SPACES

2007 ◽  
Vol 49 (3) ◽  
pp. 431-447 ◽  
Author(s):  
MASATO KIKUCHI
Keyword(s):  
Function Space ◽  
Probability Space ◽  
Banach Function Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Function Spaces ◽  
Martingale Inequalities ◽  
Necessary And Sufficient

AbstractLet X be a Banach function space over a nonatomic probability space. We investigate certain martingale inequalities in X that generalize those studied by A. M. Garsia. We give necessary and sufficient conditions on X for the inequalities to be valid.

scholarly journals SOME ASPECTS OF QUANTUM MECHANICS AND FIELD THEORY IN A LORENTZ INVARIANT NONCOMMUTATIVE SPACE

2013 ◽  
Vol 28 (07) ◽  
pp. 1350017 ◽  
Author(s):  
EVERTON M. C. ABREU ◽  
M. J. NEVES
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Spectral Representation ◽  
Free Particle ◽  
Transition Amplitude ◽  
Noncommutative Space ◽  
Moyal Product ◽  
Massive Scalar Field ◽  
Scalar Action

We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher–Fredenhagen–Roberts–Amorim (DFRA) NC background that can be considered as an alternative framework for the NC space–time of the early universe. The operators' formalism was revisited and we applied its properties to obtain an NC transition amplitude representation. Two examples of DFRA's systems were discussed, namely, the NC free particle and NC harmonic oscillator. The spectral representation of the propagator gave us the NC wave function and energy spectrum. We calculated the partition function of the NC harmonic oscillator and the distribution function. Besides, the extension to NC DFRA quantum field theory is straightforward and we used it in a massive scalar field. We had written the scalar action with self-interaction ϕ4 using the Weyl–Moyal product to obtain the propagator and vertex of this model needed to perturbation theory. It is important to emphasize from the outset, that the formalism demonstrated here will not be constructed by introducing an NC parameter in the system, as usual. It will be generated naturally from an already existing NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.

scholarly journals BMO Functions Generated by A X ℝ n Weights on Ball Banach Function Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ruimin Wu ◽  
Songbai Wang
Keyword(s):  
Linear Operator ◽  
Function Space ◽  
Maximal Function ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Bmo Functions ◽  
Littlewood Maximal Function

Let X be a ball Banach function space on ℝ n . We introduce the class of weights A X ℝ n . Assuming that the Hardy-Littlewood maximal function M is bounded on X and X ′ , we obtain that BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A X ℝ n . As a consequence, we have BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A L p · ℝ n ℝ n , where L p · ℝ n is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space X ω for any ω ∈ A X ℝ n , then the commutator b , T is bounded on X with b ∈ BMO ℝ n .

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