
ESSENTIAL SPECTRUM OF DISCRETE LAPLACIAN -
REVISITED
V. B. Kiran Kumar
Department of Mathematics, Cochin University of Science And Technology, Kochi, Kerala, (India).
E-mail:vbk@cusat.ac.in
ORCID:0000-0001-7643-4436
Reception: 21/08/2022 Acceptance: 05/09/2022 Publication: 29/12/2022
Suggested citation:
V. B. Kiran Kumar. (2022). Essential Spectrum of Discrete Laplacian - Revisited. 3C TIC. Cuadernos de desarrollo
aplicados a las TIC,11 (2), 52-59. https://doi.org/10.17993/3ctic.2022.112.52-59
https://doi.org/10.17993/3ctic.2022.112.52-59
ABSTRACT
Consider the discrete Laplacian operator
A
acting on
l2
(
Z
)
.
It is well known from the classical literature
that the essential spectrum of Ais a compact interval. In this article, we give an elementary proof for
this result, using the finite-dimensional truncations
An
of
A
. We do not rely on symbol analysis or
any infinite-dimensional arguments. Instead, we consider the eigenvalue-sequences of the truncations
An
and make use of the filtration techniques due to Arveson. Usage of such techniques to the discrete
Schrödinger operator and to the multi-dimensional settings will be interesting future problems.
KEYWORDS
Essential Spectrum, Discrete Laplacian
https://doi.org/10.17993/3ctic.2022.112.52-59
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
52