Interaction of continuous-spectrum waves with each other and with discrete-spectrum waves

1990 ◽  
Vol 24 (4) ◽  
pp. 586-592
Author(s):  
I. A. Sazonov
Keyword(s):  
Continuous Spectrum ◽  
Discrete Spectrum

scholarly journals Spectral analysis of singular Sturm-Liouville operators on time scales

2018 ◽  
Vol 72 (1) ◽  
pp. 1 ◽  
Author(s):  
Bilender P. Allahverdiev ◽  
Huseyin Tuna
Keyword(s):  
Spectral Analysis ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Time Scales ◽  
Liouville Operator ◽  
Adjoint Operator ◽  
The Self ◽  
Sturm Liouville ◽  
Liouville Operators

In this paper, we consider properties of the spectrum of a Sturm-Liouville<br />operator on time scales. We will prove that the regular symmetric<br />Sturm-Liouville operator is semi-bounded from below. We will also give some<br />conditions for the self-adjoint operator associated with the singular<br />Sturm-Liouville expression to have a discrete spectrum. Finally, we will<br />investigate the continuous spectrum of this operator.

scholarly journals The Discrete and Continuous Retardation and Relaxation Spectrum Method for Viscoelastic Characterization of Warm Mix Crumb Rubber-Modified Asphalt Mixtures

Materials ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 3723 ◽  
Author(s):  
Fei Zhang ◽  
Lan Wang ◽  
Chao Li ◽  
Yongming Xing
Keyword(s):  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Creep Compliance ◽  
Relaxation Modulus ◽  
Crumb Rubber ◽  
Asphalt Mixtures ◽  
Time Range ◽  
Modified Asphalt ◽  
Dynamic Modulus Test ◽  
Discrete And Continuous Spectrum

To study the linear viscoelastic (LVE) of crumb rubber-modified asphalt mixtures before and after the warm mix additive was added methods of obtaining the discrete and continuous spectrum are presented. Besides, the relaxation modulus and creep compliance are constructed from the discrete and continuous spectrum, respectively. The discrete spectrum of asphalt mixtures can be obtained from dynamic modulus test results according to the generalized Maxwell model (GMM) and the generalized Kelvin model (GKM). Similarly, the continuous spectrum of asphalt mixtures can be obtained from the dynamic modulus test data via the inverse integral transformation. In this paper, the test procedure for all specimens was ensured to be completed in the LVE range. The results show that the discrete spectrum and the continuous spectrum have similar shapes, but the magnitude and position of the spectrum peaks is different. The continuous spectrum can be considered as the limiting case of the discrete spectrum. The relaxation modulus and creep compliance constructed by the discrete and continuous spectrum are almost indistinguishable in the reduced time range of 10−5 s–103 s. However, there are more significant errors outside the time range, and the maximum error is up to 55%.

The eigenvalues of ∇2u + λu=0 when the boundary conditions are given on semi-infinite domains

1953 ◽  
Vol 49 (4) ◽  
pp. 668-684 ◽  
Author(s):  
D. S. Jones
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Cross Section ◽  
Radiation Condition ◽  
Finite Width ◽  
Infinite Domains ◽  
Counter Example ◽  
Submerged Cylinder

ABSTRACTThe spectrum of −∇2 (and of −∇2 + b) is investigated when the boundary conditions are given on surfaces which extend to infinity. Simple criteria are obtained for determining whether point-eigenvalues are present in the lower part of the spectrum.Semi-infinite domains which are conical at infinity are found to possess purely continuous spectra when the boundary condition is u = 0 or ∂u/∂v = 0; the radiation condition ensures a unique solution. A counter-example shows that this is not true in general for the boundary condition ∂u/∂v + σu = 0.Semi-infinite domains which are cylindrical at infinity have a continuous spectrum with a discrete spectrum embedded in it. An example is given.The results are applied to the theory of surface waves. It is shown that Ursell's ‘trapping modes’ can occur in a canal of finite width when the bed has a protrusion over a finite longth but is otherwise of uniform depth. Trapping modes can also occur when the canal contains a submerged cylinder (not necessarily small in cross-section).

scholarly journals Finite eigenfuction approximations for continuous spectrum operators

1993 ◽  
Vol 16 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Robert M. Kauffman
Keyword(s):  
Differential Equations ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Operator Norm ◽  
Abstract Theory ◽  
Eigenfunction Expansions ◽  
Multiplicity One ◽  
Finite Sums ◽  
New Formulation ◽  
Adjoint Operators

In this paper, we introduce a new formulation of the theory of continuous spectrum eigenfunction expansions for self-adjoint operators and analyze the question of when operators may be approximated in an operator norm by finite sums of multiples of eigenprojections of multiplicity one. The theory is designed for application to ordinary and partial differential equations; relationships between the abstract theory and differential equations are worked out in the paper. One motivation for the study is the question of whether these expansions are susceptible to computation on a computer, as is known to be the case for many examples in the discrete spectrum case. The point of the paper is that continuous and discrete spectrum eigenfunction expansions are treated by the same formalism; both are limits in an operator norm of finite sums.

Two Examples of Quantum Dynamical Semigroups

2011 ◽  
Vol 18 (02) ◽  
pp. 143-155
Author(s):  
Małgorzata Bodzioch ◽  
Joachim Domsta
Keyword(s):  
Density Matrix ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Reference State ◽  
Semigroup Property ◽  
Quantum Dynamical ◽  
Quantum Dynamical Semigroups ◽  
Discrete Spectra

The Hamiltonians of the considered bi-partite systems are of the form [Formula: see text] Subindex S corresponds to the observed system and R to the reservoir (the enviroment of S). Two classes of systems are distinguished: the discrete-continuous (D-C) and the continuous-continuous (C-C) models. In both cases resevoir operators MR and HR are of continuous spectrum type. In D-C models the operators HS and QS possess discrete spectra. In C-C models, the operators HS and QS are of continuous spectrum type. In each case of our examples the semigroup property for the reduced dynamics of system S is obtained under particular circumstances, depending on the diagonal of the density matrix of the reference state for R. In D-C models, due to discrete spectrum of QS, the semigroup property of the reduced dynamics of the reservoir R is shown to be impossible, unless the coupling to S is trivial.

scholarly journals A boundary value problem with a discontinuous coefficient and containing a spectral parameter in the boundary condition

1995 ◽  
Vol 18 (1) ◽  
pp. 133-140 ◽  
Author(s):  
A. A. Darwish
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Spectral Parameter ◽  
Boundary Value ◽  
Adjoint Problem ◽  
Discontinuous Coefficient ◽  
Eigenvalue Equation

A singular non-self-adjoint boundary value problem is considered. This problem has a discontinuous coefficient with a spectral parameter in the boundary condition. Some solutions of the eigenvalue equation are given. The discrete spectrum is studied and the resolvent is obtained. Formulation of the adjoint problem is deduced and hence the continuous spectrum of the considered problem is given. Furthermore, the spectrum of the adjoint problem is investigated.

Some theorems on perturbation theory. IV

1951 ◽  
Vol 210 (1100) ◽  
pp. 30-47 ◽  
Keyword(s):  
Perturbation Theory ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Real Axis ◽  
General Analysis ◽  
The Real ◽  
Perturbation Problem ◽  
Perturbed Equation ◽  
Unperturbed Equation

A general analysis is given of the perturbation problem in which the perturbed equation has a continuous spectrum, while the unperturbed equation has a discrete spectrum. It is shown that in a certain class of cases the function which determines the spectrum has, in the perturbed case, a pole just below the real axis, so that instead of a perturbed eigenvalue there is a point in the neighbourhood of which the perturbed spectrum is highly concentrated.

Transient growth associated with continuous spectra of the Batchelor vortex

2012 ◽  
Vol 697 ◽  
pp. 35-59 ◽  
Author(s):  
X. Mao ◽  
S. J. Sherwin
Keyword(s):  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Vortex Core ◽  
Azimuthal Angle ◽  
Original Position ◽  
Local Analysis ◽  
Transient Growth ◽  
Transient Effects ◽  
Potential Region ◽  
Optimal Perturbation

AbstractThe spectrum of the Batchelor vortex can be broadly split into a discrete spectrum, a potential spectrum and a free-stream spectrum where, since the last two spectra are both continuous, they can also be considered as one continuous spectrum. The discrete spectrum has been extensively studied but the continuous spectrum has received limited attention in the context of vortex flow. A local transient growth study is conducted and the contribution of the discrete spectrum and the continuous spectrum to the transient growth is separated by constructing optimal perturbations on the discrete or continuous sub-eigenspaces separately. It is found that the significant transient growth is mainly due to the non-normality of the continuous eigenmodes/spectrum whilst the discrete eigenmodes/spectrum have little contribution to the transient energy growth. A matrix-free method, which reduces to the local analysis when appropriate periodic boundary conditions are imposed, is also applied to investigate the transient growth in both a plane of constant azimuthal angle and a plane constant axial location. Previously studies by other authors have demonstrated that at zero azimuthal wavenumber the transient growth reaches infinitely large values over infinite time intervals while the optimal perturbations are located far from the vortex core. Therefore we limited our scope to small values of the time horizon so as to obtain reasonably strong transient effects stemming from physically relevant optimal perturbations. Two mechanisms of transient growth are observed: namely a redistribution of the azimuthal velocity to the azimuthal vorticity and interaction between out-of-vortex-core structures with those within the vortex core. A direct numerical simulation (DNS) of the vortex perturbed by optimal perturbations is conducted to investigate the nonlinear development of the optimal perturbations. In the azimuthally constant decomposed case, it is found that the optimal perturbation induces a string of bubble structures to be generated as a consequence of the non-orthogonality of continuous eigenmodes and the breakdown bubble is induced by viscous diffusion, while in the axially constant decomposition transient growth analysis, it is observed that the optimal perturbations associated with the continuous eigenmodes drive the vortex to vibrate around the initial vortex centre before eventually returning to its original position at larger times. This transient effect provides a mechanism for the ‘vortex meandering’ observed in previous experimental and numerical studies. These optimal perturbations associated with the continuous spectrum with out-of-vortex-core structures are observed to be activated by anisotropic inflow perturbations in the potential region.

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