1 INTRODUCTION
In this short article, we consider the discrete Laplacian operator Adefined on l2(Z), as follows:
A(x(n))=(x(n−1) + x(n+ 1)); x=(x(n)) ∈l2(Z),n∈Z.
This operator arises naturally in many physical situations. For example, when we approximate a partial
differential equation by finite differences, such bounded operators come into the picture. This operator
is widely used in image processing, particularly in edge detection problems. There are extensions of the
discrete Laplacian to various settings, such as multi-dimensional operator (on
Zn
) and Laplacian on
graphs, etc. An operator close to this example is the discrete Schrödinger operator. This operator can
be considered as a perturbation of the discrete Laplacian, defined as follows;
H(x(n))=(x(n−1) + x(n+1)+v(n)x(n)); x=(x(n)) ∈l2(Z),n∈Z.
Here the sequence v=v(n)is a bounded sequence called the potential.
It is well-known from the classical theory that the spectrum of
A
is the compact interval [
−
2
,
2]
.
In
this article, we use the filtration techniques developed by W. B. Arveson in [1] and some elementary
method to give a simple proof of this result. We plan to use such techniques in the computation of
the spectrum of the discrete Schrödinger operator. However, the spectrum of the discrete Schrödinger
operator can be very complicated, depending on the potential function. For example, if you choose the
almost Mathieu potential, the spectrum will be a Cantor-like set (The Ten-Martini Conjecture, see [2]
for, eg.).
The article is organized as follows. In the next section, we describe some essential results from [1,3]
in connection with the spectral approximation of an infinite-dimensional bounded self-adjoint operator.
In the third section, we use these techniques to give an elementary proof of the connectedness of the
essential spectrum of
A
. A possible application to the spectral computation of some special class of
discrete Schrödinger operators is mentioned at the end of this article.
2 OPERATORS IN THE ARVESON’S CLASS
"How to approximate spectra of linear operators on separable Hilbert spaces?"is a fundamental question
and was considered by many mathematicians. One of the successful methods is to use the finite-
dimensional theory in the computation of the spectrum of bounded operators in an infinite dimensional
space through an asymptotic way. In 1994
,
W.B. Arveson identified a class of operators for which the
finite-dimensional truncations are helpful in the spectral approximation [1]. We introduce this class of
operators here.
Let
A
be a bounded self-adjoint operator defined on a complex separable Hilbert space
H
and
{e1,e
2,...}
be an orthonormal basis for
H.
Consider the finite dimensional truncations of
A
, that is
An
=
PnAPn
, where
Pn
is the projection of
H
onto the span of first
n
elements
{e1,e
2,...,e
n}
of the
basis. We recall the notion of essential points and transient points introduced in [1].
Definition 1. Essential point: A real number
λ
is an essential point of
A
, if for every open set
U
containing λ, limn→∞ Nn(U)=∞,where Nn(U)is the number of eigenvalues of Anin U.
Definition 2. Transient point: A real number
λ
is a transient point of
A
if there is an open set
U
containing λ, such that sup Nn(U)with nvarying on the set of all natural number is finite.
Remark 3. Note that a number can be neither transient nor essential.
Denote Λ=
{λ∈R
;
λ
=
lim λn,λ
n∈σ
(
An
)
}
and Λ
e
as the set of all essential points. The following
spectral inclusion result for a bounded self-adjoint operator
A
is of high importance. Let
σ
(
A
)
,σ
ess
(
A
)
denote the spectrum and essential spectrum of Arespectively.
https://doi.org/10.17993/3ctic.2022.112.52-59
Theorem 4. [1] The spectrum of a bounded self-adjoint operator is a subset of the set of all limit
points of the eigenvalue sequences of its truncations. Also, the essential spectrum is a subset of the set
of all essential points. That is,
σ(A)⊆Λ⊆[m, M]and σess(A)⊆Λe.
W.B Arveson, generalized the notion of band limited matrices in [1], and achieved some useful
results in the case of some special class of operators. He used an arbitrary filtration
Hn
(an increasing
subsequence of closed subspaces with the union dense in
H
) and the sequence of orthogonal projections
onto Hnto introduce his class of operators. Here we consider only a special case.
Definition 5. The degree of a bounded operator Aon His defined by
deg(A) = sup
n≥1
rank(PnA−APn).
A Banach ∗−algebra of operators can be defined, which we call Arveson’s class, as follows.
Definition 6. Ais an operator in the Arveson’s class if
A=
∞
n=1
An,where deg(An)<∞for every n and convergence is in the
operator norm, in such a way that
∞
n=1
(1 + deg(An)1
2)∥An∥<∞
The following gives a concrete description of operators in Arveson’s class.
Theorem 7. [1] Let (
ai,j
)be the matrix representation of a bounded operator
A
, with respect to
{en}
,
and for every k∈Zlet
dk= sup
i∈Z|ai+k,i|
be the sup norm of the
kth
diagonal of (
ai,j
). Then
A
will be in Arveson’s class whenever the series
k|k|1/2dkconverges.
Remark 8. In particular, any operator whose matrix representation (
ai,j
)is band-limited, in the sense
that
ai,j
=0whenever
|i−j|
is sufficiently large, must be in Arveson’s class. Therefore, the operator
under our consideration is in Arveson’s class, as we see that its matrix representation is tridiagonal.
The following result allows us to confine our attention to essential points while looking for essential
spectral values for certain classes of operators.
Theorem 9. [1] If
A
is a bounded self-adjoint operator in the Arveson’s class, then
σess
(
A
)=Λ
e
and
every point in Λis either transient or essential.
3 SPECTRUM OF DISCRETE LAPLACIAN
Consider the discrete Laplacian operator Adefined on l2(Z), as follows:
A(x(n))=(x(n−1) + x(n+ 1)); x=(x(n)) ∈l2(Z),n∈Z.
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
54