The structure of the discrete spectrum of difference operators of first order

2000 ◽  
pp. 069-075
Author(s):  
C. COŞKUN
Keyword(s):  
Discrete Spectrum ◽  
Difference Operators ◽  
First Order

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

2010 ◽  
Vol 08 (04) ◽  
pp. 387-408 ◽  
Author(s):  
MOHAMED ALI MOUROU
Keyword(s):  
Real Line ◽  
Difference Operators ◽  
Generalized Convolution ◽  
One Dimensional ◽  
The Real ◽  
First Order ◽  
Wiener Theorem ◽  
The Fourier Transform ◽  
The One ◽  
Transmutation Operators

We consider a singular differential-difference operator Λ on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Λ and first-order regular differential-difference operators on ℝ. We exploit these transmutation operators, firstly to establish a Paley–Wiener theorem for the Fourier transform associated with Λ, and secondly to introduce a generalized convolution on ℝ tied to Λ.

On Schrödinger's factorization method for Sturm-Liouville operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 67-84 ◽  
Author(s):  
U.-W. Schmincke
Keyword(s):  
Discrete Spectrum ◽  
Liouville Operator ◽  
Differential Operators ◽  
Factorization Method ◽  
Friedrichs Extension ◽  
First Order ◽  
Sturm Liouville ◽  
Image Position ◽  
Special Case ◽  
Liouville Operators

SynopsisWe consider the Friedrichs extension A of a minimal Sturm-Liouville operator L0 and show that A admits a Schrödinger factorization, i.e. that one can find first order differential operators Bk with where the μk are suitable numbers which optimally chosen are just the lower eigenvalues of A (if any exist). With the help of this theorem we derive for the special case L0u = −u″ + q(x)u with q(x) → 0 (|x| → ∞) the inequalityσd(A) being the discrete spectrum of A. This inequality is seen to be sharp to some extent.

scholarly journals Spectral analysis of discrete dirac equation with generalized eigenparameter in boundary condition

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 6039-6054 ◽  
Author(s):  
Turhan Koprubasi ◽  
Ram Mohapatra
Keyword(s):  
Boundary Condition ◽  
Spectral Analysis ◽  
Dirac Equation ◽  
Dirac Operator ◽  
Spectral Properties ◽  
Difference Operators ◽  
Spectral Singularities ◽  
First Order ◽  
Complex Sequences

Let L denote the discrete Dirac operator generated in ?2 (N,C2) by the non-selfadjoint difference operators of first order (an+1y(2)n+1 + bny(2)n + pny(1)n = ?y(1)n, an-1y(1)n-1 + bny(1)n + qny(2)n = ?y(2)n, n ? N, (0.1) with boundary condition Xp k=0 (y(2)1?k + y(1)0 ?k)?k=0, (0.2) where (an), (bn), (pn) and (qn), n ? N are complex sequences, ?i; ?i ? C, i = 0, 1, 2,..., p and ? is a eigenparameter. We discuss the spectral properties of L and we investigate the properties of the spectrum and the principal vectors corresponding to the spectral singularities of L, if ?? n=1 |n|(|1-an| + |1+bn| + |pn| + |qn|) < ? holds.

scholarly journals An inequality for first-order differences

2005 ◽  
Vol 3 (2) ◽  
pp. 117-124
Author(s):  
Gord Sinnamon
Keyword(s):  
Difference Operators ◽  
First Order

A question about comparing norms of difference operators that was raised in [1] and presented at the Fourth ISAAC Congress is answered in the affirmative.

scholarly journals On semilocal convergence of three-step Kurchatov method under weak condition

2021 ◽  
Author(s):  
Himanshu Kumar
Keyword(s):  
Recurrence Relations ◽  
Divided Difference ◽  
Semilocal Convergence ◽  
Weak Condition ◽  
Theoretical Development ◽  
Difference Operators ◽  
Nonlinear System Of Equations ◽  
First Order ◽  
Type Integral ◽  
Weaker Conditions

AbstractThe purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the $$\omega $$ ω condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study.

scholarly journals HYERS-ULAM STABILITY OF FIRST ORDER LINEAR DIFFERENCE OPERATORS ON BANACH SPACE

2018 ◽  
Vol 14 (1) ◽  
pp. 7475-7485
Author(s):  
Arun Kumar Tripathy ◽  
Pragnya Senapati
Keyword(s):  
Banach Space ◽  
Difference Operator ◽  
Difference Operators ◽  
Ulam Stability ◽  
Order Linear ◽  
First Order ◽  
Linear Difference Operator

In this work, the Hyers-Ulam stability of first order linear difference operator TP defined by (Tpu)(n) = ∆u(n) - p(n)u(n); is studied on the Banach space X = l∞, where p(n) is a sequence of reals.

scholarly journals On sampling expansions of Kramer type

2004 ◽  
Vol 2004 (5) ◽  
pp. 371-385
Author(s):  
Anthippi Poulkou
Keyword(s):  
Boundary Value Problems ◽  
Discrete Spectrum ◽  
Differential Operators ◽  
Boundary Value ◽  
Sampling Theorem ◽  
Second Order ◽  
Joint Work ◽  
First Order ◽  
Survey Paper ◽  
Sampling Expansions

We treat some recent results concerning sampling expansions of Kramer type. The linkof the sampling theorem of Whittaker-Shannon-Kotelnikov with the Kramer sampling theorem is considered and the connection of these theorems with boundary value problems is specified. Essentially, this paper surveys certain results in the field of sampling theories and linear, ordinary, first-, and second-order boundary value problems that generate Kramer analytic kernels. The investigation of the first-order problems is tackled in a joint work with Everitt. For the second-order problems, we refer to the work of Everitt and Nasri-Roudsari in their survey paper in 1999. All these problems are represented by unbounded selfadjoint differential operators on Hilbert function spaces, with a discrete spectrum which allows the introduction of the associated Kramer analytic kernel. However, for the first-order problems, the analysis of this paper is restricted to the specification of conditions under which the associated operators have a discrete spectrum.

scholarly journals Spectrum of Linear Difference Operators and the Solvability of Nonlinear Discrete Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-27 ◽  
Author(s):  
Ruyun Ma ◽  
Youji Xu ◽  
Chenghua Gao
Keyword(s):  
Boundary Value Problems ◽  
Discrete Spectrum ◽  
Difference Equations ◽  
Nonlinear Boundary ◽  
Difference Operators ◽  
Discrete Problem ◽  
Nonlinear Boundary Value Problems ◽  
Extension Method ◽  
At Resonance ◽  
Discrete Problems

Let T∈ℕ with T>5. Let &#x1D54B;:={1,…,T}. We study the Fučik spectrum Σ of the discrete problem Δ2u(t-1)+λu+(t)-μu-(t)=0, t∈&#x1D54B;, u(0)=u(T+1)=0, where u+(t)=max⁡{u(t),0}, u-(t)=max⁡{-u(t),0}. We give an expression of Σ via the matching-extension method. We also use such discrete spectrum theory to study nonlinear boundary value problems of difference equations at resonance.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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