Spectral functions of a symmetric linear relation with a directing mapping, I

1984 ◽  
Vol 97 ◽  
pp. 165-176 ◽  
Author(s):  
H. Langer ◽  
B. Textorius
Keyword(s):  
Spectral Function ◽  
Differential System ◽  
Linear Relation ◽  
Spectral Functions ◽  
First Order ◽  
Symmetric Linear Relation

SynopsisFor a symmetric linear relation S with a directing mapping, the notion of a spectral function is defined by means of a Bessel–Parseval inequality, and a description of all such spectral functions is given. As an application, we describe the set of all spectral functions of a canonical regular first order differential system.

Spectral functions of a symmetric linear relation with a directing mapping, II

1985 ◽  
Vol 101 (1-2) ◽  
pp. 111-124 ◽  
Author(s):  
H. Langer ◽  
B. Textorius
Keyword(s):  
Moment Problem ◽  
Linear Relation ◽  
Spectral Functions ◽  
Differential Systems ◽  
Symmetric Linear Relation

SynopsisThe results of part I (see [5]) are applied to pairs of formally symmetric differential expressions, to Hermitian differential systems and to a reduced operator moment problem.

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

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.

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.

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.

scholarly journals Spectral asymptotics for canonical systems

2018 ◽  
Vol 2018 (736) ◽  
pp. 285-315 ◽  
Author(s):  
Jonathan Eckhardt ◽  
Aleksey Kostenko ◽  
Gerald Teschl
Keyword(s):  
High Energy ◽  
Spectral Asymptotics ◽  
Spectral Functions ◽  
Open Problems ◽  
New Approach ◽  
Canonical Systems ◽  
First Order ◽  
Continuity Properties ◽  
High Energy Behavior ◽  
Energy Behavior

AbstractBased on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of Weyl–Titchmarsh and spectral functions of{2\times 2}first order canonical systems. Our results improve several classical results and solve open problems posed by previous authors. Furthermore, they are applied to radial Dirac and radial Schrödinger operators as well as to Krein strings and generalized indefinite strings.

Center Problems and Limit Cycle Bifurcations in a Class of Quasi-Homogeneous Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550135 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han ◽  
Yong Wang
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System ◽  
Generalized Polynomial ◽  
Necessary And Sufficient

In this paper, we first classify all centers of a class of quasi-homogeneous polynomial differential systems of degree 5. Then we extend this kind of systems to a generalized polynomial differential system and provide the necessary and sufficient conditions to have a center at the origin. Furthermore, we study the Poincaré bifurcation for its perturbed system as it has a center at the origin, find the Poincaré cyclicity up to first order of ε.

scholarly journals Spectral Function of a Boson Ladder in an Artificial Gauge Field

2020 ◽  
Vol 5 (1) ◽  
pp. 15 ◽  
Author(s):  
Roberta Citro ◽  
Stefania De Palo ◽  
Nicolas Victorin ◽  
Anna Minguzzi ◽  
Edmond Orignac
Keyword(s):  
Magnetic Field ◽  
Magnetic Flux ◽  
Spectral Function ◽  
Weak Interactions ◽  
Spectral Weight ◽  
Ultracold Atom ◽  
Spectral Functions ◽  
Vortex Phase ◽  
Bogoliubov Theory ◽  
Radio Frequency Spectroscopy

We calculate the spectral function of a boson ladder in an artificial magnetic field by means of analytic approaches based on bosonization and Bogoliubov theory. We discuss the evolution of the spectral function at increasing effective magnetic flux, from the Meissner to the Vortex phase, focussing on the effects of incommensurations in momentum space. At low flux, in the Meissner phase, the spectral function displays both a gapless branch and a gapped one, while at higher flux, in the Vortex phase, the spectral function displays two gapless branches and the spectral weight is shifted at a wavevector associated to the underlying vortex spatial structure, which can indicate a supersolid-like behavior. While the Bogoliubov theory, valid at weak interactions, predicts sharp delta-like features in the spectral function, at stronger interactions we find power-law broadening of the spectral functions due to quantum fluctuations as well as additional spectral weight at higher momenta due to backscattering and incommensuration effects. These features could be accessed in ultracold atom experiments using radio-frequency spectroscopy techniques.

scholarly journals Evolution of Spectral Functions in Doped Transition Metal Oxides

1997 ◽  
Vol 11 (32) ◽  
pp. 3849-3857 ◽  
Author(s):  
H. Kajueter ◽  
G. Kotliar ◽  
D. D. Sarma ◽  
S. R. Barman
Keyword(s):  
Transition Metal ◽  
Metal Oxides ◽  
Spectral Function ◽  
Transition Metal Oxides ◽  
Mean Field Theory ◽  
Mean Field ◽  
Correlated Electrons ◽  
Infinite Dimensions ◽  
Spectral Functions ◽  
Early Transition Metal

We discuss the experimental photoemission and inverse photoemission of early transition metal oxides, in the light of the dynamical mean field theory of correlated electrons which becomes exact in the limit of infinite dimensions. We argue that a comprehensive description of the experimental data requires spatial inhomogeneities and present a calculation of the evolution of the spectral function in an inhomogeneities and present a calculation of the evolution of the spectral function in an inhomogenous system with various degrees of inhomogeneity. We also point out that comparison of experimental results and large d calculations require that the degree of correlation and disorder is larger in the surface than in the bulk.

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