The asymptotic behavior of the discrete spectrum in the gaps of the continuous spectrum of a perturbed hill operator

1991 ◽  
Vol 25 (2) ◽  
pp. 162-164
Author(s):  
A. V. Sobolev
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Hill Operator

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.

High Frequency Vibrations in a Stiff Problem

1997 ◽  
Vol 07 (02) ◽  
pp. 291-311 ◽  
Author(s):  
Miguel Lobo ◽  
Eugenia Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Spectral Problem ◽  
Continuous Spectrum ◽  
High Frequency ◽  
The Other ◽  
Stiff Problem ◽  
Eigenvalues And Eigenfunctions ◽  
High Frequencies ◽  
Stiffness Constant ◽  
High Frequency Vibrations

The stiff problem here considered models the vibrations of a body consisting of two materials, one of them very stiff with respect to the other. We study the asymptotic behavior of the eigenvalues and eigenfunctions of the corresponding spectral problem, when the stiffness constant of only one of the materials tends to 0. We show that the associated operator has a discrete spectrum "converging", in a certain sense, towards a continuous spectrum in [0,∞) corresponding to an operator. We also provide information on the structure of the eigenfunctions associated with the high frequencies.

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%.

scholarly journals Asymptotics for Helmholtz and Maxwell Solutions in 3-D Open Waveguides

2012 ◽  
Vol 11 (2) ◽  
pp. 629-646 ◽  
Author(s):  
Carlos Jerez-Hanckes ◽  
Jean-Claude Nédélec
Keyword(s):  
Asymptotic Behavior ◽  
Discrete Spectrum ◽  
Electromagnetic Fields ◽  
Three Dimensions ◽  
Guided Modes ◽  
Open Waveguides ◽  
Radiation Conditions

AbstractWe extend classic Sommerfeld and Silver-Müller radiation conditions for bounded scatterers to acoustic and electromagnetic fields propagating over three isotropic homogeneous layers in three dimensions. If X= (x1,x2,x3)ϵℝ3, with x3 denoting the direction orthogonal to the layers, standard conditions only hold for the outer layers in the region ∣x3∣ > ∣∣x∣γ, for γϵ(1/4,1/2) and x large. For ∣x3∣ < ∣∣x∣∣γ and inside the slab, asymptotic behavior depends on the presence of surface or guided modes given by the discrete spectrum of the associated operator.

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).

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