Exponential behaviour of eigenfunctions and gaps in the essential spectrum

1980 ◽  
pp. 1-24
Author(s):  
F. V. Atkinson
Keyword(s):  
Essential Spectrum ◽  
Exponential Behaviour

On the exponential behaviour of eigenfunctions and the essential spectrum of differential operators

1982 ◽  
Vol 92 (3-4) ◽  
pp. 271-300
Author(s):  
F. V. Atkinson ◽  
W. D. Evans
Keyword(s):  
Differential Equation ◽  
Linear Operator ◽  
Essential Spectrum ◽  
General Result ◽  
Differential Operators ◽  
Exponential Behaviour ◽  
Image Position ◽  
Complex Valued

SynopsisThe paper deals with the differential equationon [ 0, ∞) Where λ>0 and the coefficients qm are complex-valued with qn continuous and non-zero, w is positive and continuous and qm for m = 0, 1,…, n − 1. In the first part of the paper the exponential behaviour of any solution of (*) is given in terms of a function ρ(λ) which is roughly the distance of λ from the essential spectrum of a closed, densely denned linear operator T generated by T+ in L2(0, ∞ w). Next, estimates are obtained for the solutions in terms of the coefficients in (*). When the latter results are compared with the estimates established previously in terms of ρ(λ), bounds for ρ(λ) are obtained. From the general result there are two kinds of consequences. In the first, criteria for ρ(λ) = 0 for all All λ > 0 are obtained; this means that [0, ∞) lies in the essential spectrum of T in appropriate circumstances. The second type of consequence concerns bounds of the form ρ(λ) = O(λr) for λ → ∞ and r<1.

New branches of the essential spectrum of a 2 × 2 operator matrix

2020 ◽  
Vol 2020 (2) ◽  
pp. 44-51
Author(s):  
E.B. Dilmurodov
Keyword(s):  
Essential Spectrum ◽  
Operator Matrix

STABILITY OF STEADY STATE AND TRAVELING WAVES SOLUTIONS IN COUPLED MAP LATTICES

2008 ◽  
Vol 18 (01) ◽  
pp. 219-225 ◽  
Author(s):  
DANIEL TURZÍK ◽  
MIROSLAVA DUBCOVÁ
Keyword(s):  
Steady State ◽  
Essential Spectrum ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Linear Operators ◽  
Coupled Map Lattices ◽  
Basic Tool ◽  
Wave Solutions ◽  
The Stability

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

scholarly journals On Gromov's theorem andL 2-Hodge decomposition

2004 ◽  
Vol 2004 (1) ◽  
pp. 25-44 ◽  
Author(s):  
Fu-Zhou Gong ◽  
Feng-Yu Wang
Keyword(s):  
Essential Spectrum ◽  
Vector Bundles ◽  
Functional Inequality ◽  
Upper Bounds ◽  
Hodge Decomposition ◽  
Laplace Type

Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the correspondingL 2-harmonic sections. In particular, some known results concerning Gromov's theorem and theL 2-Hodge decomposition are considerably improved.

Marangoni instability of a thin liquid film heated from below by a local heat source

2003 ◽  
Vol 475 ◽  
pp. 377-408 ◽  
Author(s):  
SERAFIM KALLIADASIS ◽  
ALLA KIYASHKO ◽  
E. A. DEMEKHIN
Keyword(s):  
Free Surface ◽  
Liquid Film ◽  
Discrete Spectrum ◽  
Essential Spectrum ◽  
Marangoni Number ◽  
Thin Liquid Film ◽  
Nonlinear Partial Differential Equations ◽  
Spanwise Direction ◽  
Surface Boundary ◽  
Integral Boundary

We consider the motion of a liquid film falling down a heated planar substrate. Using the integral-boundary-layer approximation of the Navier–Stokes/energy equations and free-surface boundary conditions, it is shown that the problem is governed by two coupled nonlinear partial differential equations for the evolution of the local film height and temperature distribution in time and space. Two-dimensional steady-state solutions of these equations are reported for different values of the governing dimensionless groups. Our computations demonstrate that the free surface develops a bump in the region where the wall temperature gradient is positive. We analyse the linear stability of this bump with respect to disturbances in the spanwise direction. We show that the operator of the linearized system has both a discrete and an essential spectrum. The discrete spectrum bifurcates from resonance poles at certain values of the wavenumber for the disturbances in the transverse direction. The essential spectrum is always stable while part of the discrete spectrum becomes unstable for values of the Marangoni number larger than a critical value. Above this critical Marangoni number the growth rate curve as a function of wavenumber has a finite band of unstable modes which increases as the Marangoni number increases.

scholarly journals The Weyl essential spectrum

1971 ◽  
Vol 28 (2) ◽  
pp. 531-531 ◽  
Author(s):  
Richard Bouldin
Keyword(s):  
Essential Spectrum
Sign in / Sign up

Export Citation Format

Share Document