Theorems on the coincidence of the spectra of pseudodifferential almost-periodic operators in the spaces L2(Rn) and B2(Rn)

1976 ◽  
Vol 17 (1) ◽  
pp. 158-170 ◽  
Author(s):  
M. A. Shubin
Keyword(s):  
Almost Periodic ◽  
Periodic Operators

scholarly journals The spectrum of a class of almost periodic operators

2003 ◽  
Vol 2003 (36) ◽  
pp. 2277-2301 ◽  
Author(s):  
Norbert Riedel
Keyword(s):  
Almost Periodic ◽  
Cantor Spectrum ◽  
Periodic Operators

For almost Mathieu operators, it is shown that the occurrence of Cantor spectrum and the existence, for every point in the spectrum and suitable phase parameters, of at least one localized eigenfunction which decays exponentially are inconsistent properties.

Bloch wave homogenisation of quasiperiodic media

2020 ◽  
pp. 1-21
Author(s):  
SISTA SIVAJI GANESH ◽  
VIVEK TEWARY
Keyword(s):  
Bloch Wave ◽  
Periodic Operator ◽  
Almost Periodic ◽  
Periodic Media ◽  
Direct Integral ◽  
Bloch Waves ◽  
Direct Integral Decomposition ◽  
Quasiperiodic Media ◽  
Integral Decomposition ◽  
Periodic Operators

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

scholarly journals DISCRETE ALMOST PERIODIC OPERATORS

Cubo (Temuco) ◽  
2013 ◽  
Vol 15 (1) ◽  
pp. 171-189 ◽  
Author(s):  
Alexander Pankov
Keyword(s):  
Almost Periodic ◽  
Periodic Operators
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