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

acting on

(

)

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

. We do not rely on symbol analysis or

any inﬁnite-dimensional arguments. Instead, we consider the eigenvalue-sequences of the truncations

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

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

acting on

(

)

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

. We do not rely on symbol analysis or

any inﬁnite-dimensional arguments. Instead, we consider the eigenvalue-sequences of the truncations

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

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

) 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

is the compact interval [

−

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 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

be a bounded self-adjoint operator deﬁned on a complex separable Hilbert space

and

{e1,e

2,...}

be an orthonormal basis for

Consider the ﬁnite dimensional truncations of

, that is

PnAPn

, where

is the projection of

onto the span of ﬁrst

elements

{e1,e

2,...,e

of the

basis. We recall the notion of essential points and transient points introduced in [1].

Deﬁnition 1. Essential point: A real number

is an essential point of

, if for every open set

containing λ, limn→∞ Nn(U)=∞,where Nn(U)is the number of eigenvalues of Anin U.

Deﬁnition 2. Transient point: A real number

is a transient point of

if there is an open set

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

;

lim λn,λ

n∈σ

(

)

}

and Λ

as the set of all essential points. The following

spectral inclusion result for a bounded self-adjoint operator

is of high importance. Let

(

)

,σ

ess

(

)

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

(an increasing

subsequence of closed subspaces with the union dense in

) 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

∞



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

, 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

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

is a bounded self-adjoint operator in the Arveson’s class, then

σess

(

)=Λ

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

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

) 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

is the compact interval [

−

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 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

be a bounded self-adjoint operator deﬁned on a complex separable Hilbert space

and

{e1,e

2,...}

be an orthonormal basis for

Consider the ﬁnite dimensional truncations of

, that is

PnAPn

, where

is the projection of

onto the span of ﬁrst

elements

{e1,e

2,...,e

of the

basis. We recall the notion of essential points and transient points introduced in [1].

Deﬁnition 1. Essential point: A real number

is an essential point of

, if for every open set

containing λ, limn→∞ Nn(U)=∞,where Nn(U)is the number of eigenvalues of Anin U.

Deﬁnition 2. Transient point: A real number

is a transient point of

if there is an open set

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

;

lim λn,λ

n∈σ

(

)

}

and Λ

as the set of all essential points. The following

spectral inclusion result for a bounded self-adjoint operator

is of high importance. Let

(

)

,σ

ess

(

)

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

(an increasing

subsequence of closed subspaces with the union dense in

) 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

∞



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

, 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

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

is a bounded self-adjoint operator in the Arveson’s class, then

σess

(

)=Λ

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

If we use the standard orthonormal basis on

(

), the truncations

will have the following matrix

representations:

An=







0100 .

10100 .

010100 .

0010100 .

.0010100

.001010

.00101

.0010







Now we recollect some properties of the discrete Laplacian operator below.

Lemma 1. λ∈σess (A)if and only if −λ∈σess (A).

Proof. Notice that this operator

is in Arveson’s class, introduced in the last section. Therefore, we

have

λ∈σess (A)

if and only if

is an essential point. That is exactly when

Nn(U)→∞

for every

neighborhood

λ.

The characteristic polynomials of

are

Pn(z)

zn−an−2zn−2

...±

when

is even and

Pn(z)

−zn

an−2zn−2−...±a1z

, when

is odd. Here the coeﬃcients can be

computed as follows. ak=(n−k+2)(n−k+4)...(n+k)

2k.k!

Therefore the eigenvalues of

are distributed symmetrically on both sides of 0in the interval

[−2,2] .

Hence the number of truncated eigenvalues in any neighbourhood of

−λ

and

are the

same if the neighbourhoods are of the same length. We can conclude that

λ∈σess (A)

if and only if

−λ∈σess (A).

Lemma 2. The operator norm ∥A∥=2and ±2∈σess (A).

Proof. For every x∈l2(Z), we have

∥Ax∥2=

∞



−∞

(x(n−1) + x(n+ 1))2=

∞



−∞ (x(n−1))2+(x(n+ 1))2+2x(n−1)x(n+ 1)≤4∥x∥2.

Therefore, ∥A∥≤2.

To prove the equality, consider the sequence

�...0,0,0, ... 1

n,1

n... 1

n,0,0,0...,

where

repeats

times and all other entries are 0

Then

has norm 1and

∥Axn∥

increases to 2

Hence

∥A∥

Since

is a bounded self-adjoint operator, either

∥A∥

−∥A∥

is always a spectral value. That is 2or

−2is in the spectrum, σ(A). However, they are not eigenvalues of A, as we see below.

2is an eigenvalue of

, then 4is an eigenvalue of

where

is deﬁned by

B(en)

en−2

en+2.

(Observe that

is deﬁned as

A2(en)

en−2

en+2

). This will imply

that 2is eigenvalue of

, for some nonzero

then

(

+2)+

(

n−

2) = 2

(

), for all

Let

(

p

for some

and

(

N−

2) =

Then

(

)=(

+ 1)

p−kq

for every

. Such

an element xwill not be in l2(Z).

Hence

2is an essential spectral value. By Lemma 1, both

2are in the essential spectrum,

σess (A)

Therefore both 2 and -2 are in σess (A).

Theorem 10. The spectral gaps of

if they exist, will appear symmetrically with respect to the origin.

That means corresponding to each spectral gap on the positive real axis, there exists a spectral gap on

the negative real axis. In particular, Acannot have an odd number of spectral gaps.

https://doi.org/10.17993/3ctic.2022.112.52-59

Proof. We noticed that

where

is deﬁned by

B(en)

en−2

en+2.

It is worthwhile

to notice further that ∥B∥=2.This follows easily from the following arguments.

∥Bx∥2=

∞



−∞

(x(n−2) + x(n+ 2))2=

∞



−∞ (x(n−2))2+(x(n+ 2))2+2x(n−2)x(n+ 2)≤4∥x∥2.

Therefore we have

∥B∥≤

Now consider the sequence

...0,0,0, ... 1

n,1

n... 1

n,0,0,0...

where

repeats

times and all other entries 0

has norm 1and

∥Bxn∥

increases to 2

Hence

∥B∥

Also as in the case of

the truncated eigenvalues are distributed symmetrically on both sides of 0

that

−

2and 2are in the essential spectrum. Also, since

this implies that 0is an essential

spectral value of

, and hence of

. Hence any spectral gap of

can occur either to the right or left

side of 0

By Lemma 1, each spectral gap on the right side of 0will also give a spectral gap on the left

side. Hence the proof.

Theorem 11. The essential spectrum of

is connected. The spectrum and the essential spectrum

coincide with the compact interval [−2,2].

Proof. First, we show that

has no eigenvalues. This will imply that the spectrum and essential

spectrum coincide, as the essential spectrum consists of discrete eigenvalues of ﬁnite multiplicity. If

Ax =λx, for some nonzero x, then

x(n+1)+x(n−1) = λx(n),for all n.

Let

(

p

for some

and

(

N−

1) =

Then a recursive argument similar to that in the

proof of Lemma 2 will show that such a vector will not lie in l2(Z).

By Theorem 10, spectral gaps can occur symmetrically to the origin. Hence it suﬃces to show that

there is no spectral gap to the right side of the origin. We consider each possible case for the existence

of a spectral gap. We rule them out one by one. First consider the case when the spectral gap is of the

form

(a, 2) ,

with 0

≤a≤

1. In this case, since the interval

(a, 2)

contains no essential points (as the

essential spectrum coincides with the set of all essential points), it will contain at most

eigenvalues of

truncations for inﬁnitely many

. Let

λn1,λ

n2,λ

n3,...λ

be those eigenvalues. From the expression

of characteristic polynomials, it is evident that the determinant of

A′

is either 0or

1. Since the

eigenvalues are distributed symmetrically to both sides of 0, we have the product of positive eigenvalues

equal 1for neven. Let sK:= K

i=1 λni. Then 1

sK>1

2K,since λni <2for i=1,2,...K.

But since 0is in the essential spectrum, it is an essential point, and we can ﬁnd an

such that

the interval

0,1

2

contains at least

+1eigenvalues of

for every

n≥N

. For such an

n≥N,

let

αn1,α

n2,α

n3,...α

nN−Kbe the eigenvalues of An, in (0,a). Therefore we have,

2K<1

N−K



i=1

αni <

K+1



i=1

αni <1

2K+1 <1

2K.

The ﬁrst equality holds since the product of eigenvalues is 1, and the consequent inequality is because

a≤

1(each additional

αni

we multiply will be a positive number below 1. Hence, the product will

satisfy this inequality). This contradiction leads to the fact that

(a, 2) ,

with 0

≤a≤

1is not a spectral

gap.

Now we see that

(a, 2) ,

with

1cannot be a spectral gap. For if

(a, 2) ,a>

1is a gap, then it

will contain at most

eigenvalues of truncations for inﬁnitely many

. Let

λn1,λ

n2,λ

n3,...λ

those eigenvalues. As in the above case, let

K

i=1 λni

. Then

sK>1

Find an

such that the

interval

0,1

2aN

contains at least

+1eigenvalues of

. Now let

αn1,α

n2,α

n3,...α

nN−K

be the

eigenvalues of AN, in (0,a). Therefore we have

N−K



i=1

αni <1

2aNK+1

aN−(2K+1) <1

2K+1

2K.

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

If we use the standard orthonormal basis on

(

), the truncations

will have the following matrix

representations:

An=







0100 .

10100 .

010100 .

0010100 .

.0010100

.001010

.00101

.0010







Now we recollect some properties of the discrete Laplacian operator below.

Lemma 1. λ∈σess (A)if and only if −λ∈σess (A).

Proof. Notice that this operator

is in Arveson’s class, introduced in the last section. Therefore, we

have

λ∈σess (A)

if and only if

is an essential point. That is exactly when

Nn(U)→∞

for every

neighborhood

λ.

The characteristic polynomials of

are

Pn(z)

zn−an−2zn−2

...±

when

is even and

Pn(z)

−zn

an−2zn−2−...±a1z

, when

is odd. Here the coeﬃcients can be

computed as follows. ak=(n−k+2)(n−k+4)...(n+k)

2k.k!

Therefore the eigenvalues of

are distributed symmetrically on both sides of 0in the interval

[−2,2] .

Hence the number of truncated eigenvalues in any neighbourhood of

−λ

and

are the

same if the neighbourhoods are of the same length. We can conclude that

λ∈σess (A)

if and only if

−λ∈σess (A).

Lemma 2. The operator norm ∥A∥=2and ±2∈σess (A).

Proof. For every x∈l2(Z), we have

∥Ax∥2=

∞



−∞

(x(n−1) + x(n+ 1))2=

∞



−∞ (x(n−1))2+(x(n+ 1))2+2x(n−1)x(n+ 1)≤4∥x∥2.

Therefore, ∥A∥≤2.

To prove the equality, consider the sequence

�...0,0,0, ... 1

n,1

n... 1

n,0,0,0...,

where

repeats

times and all other entries are 0

Then

has norm 1and

∥Axn∥

increases to 2

Hence

∥A∥

Since

is a bounded self-adjoint operator, either

∥A∥

−∥A∥

is always a spectral value. That is 2or

−2is in the spectrum, σ(A). However, they are not eigenvalues of A, as we see below.

2is an eigenvalue of

, then 4is an eigenvalue of

where

is deﬁned by

B(en)

en−2

en+2.

(Observe that

is deﬁned as

A2(en)

en−2

en+2

). This will imply

that 2is eigenvalue of

, for some nonzero

then

(

+2)+

(

n−

2) = 2

(

), for all

Let

(

p

for some

and

(

N−

2) =

Then

(

)=(

+ 1)

p−kq

for every

. Such

an element xwill not be in l2(Z).

Hence

2is an essential spectral value. By Lemma 1, both

2are in the essential spectrum,

σess (A)

Therefore both 2 and -2 are in σess (A).

Theorem 10. The spectral gaps of

if they exist, will appear symmetrically with respect to the origin.

That means corresponding to each spectral gap on the positive real axis, there exists a spectral gap on

the negative real axis. In particular, Acannot have an odd number of spectral gaps.

https://doi.org/10.17993/3ctic.2022.112.52-59

Proof. We noticed that

where

is deﬁned by

B(en)

en−2

en+2.

It is worthwhile

to notice further that ∥B∥=2.This follows easily from the following arguments.

∥Bx∥2=

∞



−∞

(x(n−2) + x(n+ 2))2=

∞



−∞ (x(n−2))2+(x(n+ 2))2+2x(n−2)x(n+ 2)≤4∥x∥2.

Therefore we have

∥B∥≤

Now consider the sequence

...0,0,0, ... 1

n,1

n... 1

n,0,0,0...

where

repeats

times and all other entries 0

has norm 1and

∥Bxn∥

increases to 2

Hence

∥B∥

Also as in the case of

the truncated eigenvalues are distributed symmetrically on both sides of 0

that

−

2and 2are in the essential spectrum. Also, since

this implies that 0is an essential

spectral value of

, and hence of

. Hence any spectral gap of

can occur either to the right or left

side of 0

By Lemma 1, each spectral gap on the right side of 0will also give a spectral gap on the left

side. Hence the proof.

Theorem 11. The essential spectrum of

is connected. The spectrum and the essential spectrum

coincide with the compact interval [−2,2].

Proof. First, we show that

has no eigenvalues. This will imply that the spectrum and essential

spectrum coincide, as the essential spectrum consists of discrete eigenvalues of ﬁnite multiplicity. If

Ax =λx, for some nonzero x, then

x(n+1)+x(n−1) = λx(n),for all n.

Let

(

p

for some

and

(

N−

1) =

Then a recursive argument similar to that in the

proof of Lemma 2 will show that such a vector will not lie in l2(Z).

By Theorem 10, spectral gaps can occur symmetrically to the origin. Hence it suﬃces to show that

there is no spectral gap to the right side of the origin. We consider each possible case for the existence

of a spectral gap. We rule them out one by one. First consider the case when the spectral gap is of the

form

(a, 2) ,

with 0

≤a≤

1. In this case, since the interval

(a, 2)

contains no essential points (as the

essential spectrum coincides with the set of all essential points), it will contain at most

eigenvalues of

truncations for inﬁnitely many

. Let

λn1,λ

n2,λ

n3,...λ

be those eigenvalues. From the expression

of characteristic polynomials, it is evident that the determinant of

A′

is either 0or

1. Since the

eigenvalues are distributed symmetrically to both sides of 0, we have the product of positive eigenvalues

equal 1for neven. Let sK:= K

i=1 λni. Then 1

sK>1

2K,since λni <2for i=1,2,...K.

But since 0is in the essential spectrum, it is an essential point, and we can ﬁnd an

such that

the interval

0,1

2

contains at least

+1eigenvalues of

for every

n≥N

. For such an

n≥N,

let

αn1,α

n2,α

n3,...α

nN−Kbe the eigenvalues of An, in (0,a). Therefore we have,

2K<1

N−K



i=1

αni <

K+1



i=1

αni <1

2K+1 <1

2K.

The ﬁrst equality holds since the product of eigenvalues is 1, and the consequent inequality is because

a≤

1(each additional

αni

we multiply will be a positive number below 1. Hence, the product will

satisfy this inequality). This contradiction leads to the fact that

(a, 2) ,

with 0

≤a≤

1is not a spectral

gap.

Now we see that

(a, 2) ,

with

1cannot be a spectral gap. For if

(a, 2) ,a>

1is a gap, then it

will contain at most

eigenvalues of truncations for inﬁnitely many

. Let

λn1,λ

n2,λ

n3,...λ

those eigenvalues. As in the above case, let

K

i=1 λni

. Then

sK>1

Find an

such that the

interval

0,1

2aN

contains at least

+1eigenvalues of

. Now let

αn1,α

n2,α

n3,...α

nN−K

be the

eigenvalues of AN, in (0,a). Therefore we have

N−K



i=1

αni <1

2aNK+1

aN−(2K+1) <1

2K+1

2K.

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

The inequality is a consequence of

1. This contradiction leads to the fact that

(a, 2) ,a>

1is not

a gap.

Hence we have seen that there cannot have a spectral gap of the form (

2), with

being a non-

negative real number. The number 2does not play any role in the proof. We can easily imitate the

proof techniques for intervals of the form (a, b), with 0≤a<b≤2.

Remark 12. We can have a diﬀerent and straight forward argument to show that

(a, 2)

cannot be a

spectral gap. However, this method cannot be extended for arbitrary intervals (

a, b

), with 0

≤a<b≤

For if

(a, 2)

is a gap, then 2will be an isolated point in the essential spectrum. Since the interval

(a, 2)

contains at most

eigenvalues of truncations, but 2is an essential point, we need 2is an eigenvalue of

for large values of

with multiplicity increases to inﬁnity. Nevertheless, 2is not an eigenvalue of

Anfor any n. For if







101

...

101































then

x2=2x1

x3=3x1

x4=4x1, ...x

n−1=(n−1)x1,x

n=nx1

Also xn−1=2xn=2nx1

But this will hold only when

=0which makes

=0and hence 2 is not an eigenvalue. Hence we

conclude that (a, 2) is not a gap.

Remark 13. The eigenvalues of the matrices

are explicitly calculated to be 2

cos

(

πk/n +1

)

1,2,3...n. We may arrive at some conclusions from that information also.

Remark 14. An important question is whether we can approximate the eigenvalues of

using the

eigenvalues of truncation. Since the operator we considered has no eigenvalues, it is interesting to see

from the truncations itself whether the operator has an eigenvalue or not. In general, we observe the

following; If

lim λn,λ

n∈σ(An)

and if the sequence of eigenvectors

corresponding to

λn

, is

Cauchy in H,then λis an eigenvalue of A. This can easily be seen as follows.

Let

converges to some

H.λ

lim λn,x

lim xn

together imply

λx

lim λnxn

Also

lim Anxn

. Hence for any

ϵ>

there is an

such that

∥λ−λNxN∥<ε

2,∥Ax −ANxN∥<ε

Hence for any ϵ>0,∥Ax −λx∥<ε. That is λis an eigenvalue of A.

4 CONCLUDING REMARKS

We used only elementary tools and the ﬁltration techniques due to Arveson to prove the connectedness

of the essential spectrum. When we consider the Discrete Schrödinger operator Hdeﬁned by

H(x(n))=(x(n−1) + x(n+1)+v(n)x(n)); x=(x(n)) ∈l2(Z),n∈Z,

with the potential sequence

(

)being periodic, there will be spectral gaps unless when

is constant

(see [4,5] for example). However, if we write the matrix representation with respect to the standard

orthonormal basis, it is tridiagonal; hence, Arveson’s techniques are available. Here the spectral analysis

depends on the nature of the potential. The spectral gap issues of such operators were studied with the

linear algebraic techniques in [6]. The spectral gap issues of arbitrary bounded self-adjoint operators

can be found in the literature (see [7,8] for example).

https://doi.org/10.17993/3ctic.2022.112.52-59

Another interesting point is to carry over such techniques to the multi-dimensional case by replacing

Zby Zn.

ACKNOWLEDGMENT

V.B. Kiran Kumar wishes to thank KSCSTE, Kerala, for ﬁnancial support via the YSA-Research

project.

REFERENCES

[1]

W.B. Arveson, (1994).

C∗

- algebras and numerical linear algebra. J. Funct. Anal. 122 , no.2, pp.

333–360.

[2]

A. Avila, and S. Jitomirskaya (2009). The Ten Martini Problem Ann. of Math. 170 , no.1, pp.

303–342.

[3]

A. Böttcher,A.V. Chithra, and M.N.N. Namboodiri (2001). Approximation of approximation

numbers by truncation Integral Equations Operator Theory 39 , pp. 387–395.

[4]

L. Golinskii,V. B. Kiran Kumar,M.N.N. Namboodiri, and S. Serra-Capizzano, (2013).

A note on a discrete version of Borg’s Theorem via Toeplitz-Laurent operators with matrix-valued

symbols Boll. Unione Mat. Ital. 39, no.1 , pp. 205–218.

[5]

V. B. Kiran Kumar,M.N.N. Namboodiri, and S. Serra-Capizzano, (2014). Perturbation of

operators and approximation of spectrum Proc. Indian Acad. Sci. Math. Sci. 124 (2014), no. 2, ,

pp. 205–224.

[6]

V. B. Kiran Kumar(2015). Truncation Method for Random Bounded Self-adjoint Operators

Banach J. Math. Anal. 9 no.3, pp. 98–113.

[7]

M.N.N. Namboodiri(2002). Truncation method for Operators with disconnected essential spec-

trum Proc.Indian Acad.Sci.(MathSci) 112 , pp. 189–193.

[8] M.N.N. Namboodiri(2005). Theory of spectral gaps- A short survey J.Analysis 12 , pp. 1–8.

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

The inequality is a consequence of

1. This contradiction leads to the fact that

(a, 2) ,a>

1is not

a gap.

Hence we have seen that there cannot have a spectral gap of the form (

2), with

being a non-

negative real number. The number 2does not play any role in the proof. We can easily imitate the

proof techniques for intervals of the form (a, b), with 0≤a<b≤2.

Remark 12. We can have a diﬀerent and straight forward argument to show that

(a, 2)

cannot be a

spectral gap. However, this method cannot be extended for arbitrary intervals (

a, b

), with 0

≤a<b≤

For if

(a, 2)

is a gap, then 2will be an isolated point in the essential spectrum. Since the interval

(a, 2)

contains at most

eigenvalues of truncations, but 2is an essential point, we need 2is an eigenvalue of

for large values of

with multiplicity increases to inﬁnity. Nevertheless, 2is not an eigenvalue of

Anfor any n. For if







101

...

101































then

x2=2x1

x3=3x1

x4=4x1, ...x

n−1=(n−1)x1,x

n=nx1

Also xn−1=2xn=2nx1

But this will hold only when

=0which makes

=0and hence 2 is not an eigenvalue. Hence we

conclude that (a, 2) is not a gap.

Remark 13. The eigenvalues of the matrices

are explicitly calculated to be 2

cos

(

πk/n +1

)

1,2,3...n. We may arrive at some conclusions from that information also.

Remark 14. An important question is whether we can approximate the eigenvalues of

using the

eigenvalues of truncation. Since the operator we considered has no eigenvalues, it is interesting to see

from the truncations itself whether the operator has an eigenvalue or not. In general, we observe the

following; If

lim λn,λ

n∈σ(An)

and if the sequence of eigenvectors

corresponding to

λn

, is

Cauchy in H,then λis an eigenvalue of A. This can easily be seen as follows.

Let

converges to some

H.λ

lim λn,x

lim xn

together imply

λx

lim λnxn

Also

lim Anxn

. Hence for any

ϵ>

there is an

such that

∥λ−λNxN∥<ε

2,∥Ax −ANxN∥<ε

Hence for any ϵ>0,∥Ax −λx∥<ε. That is λis an eigenvalue of A.

4 CONCLUDING REMARKS

We used only elementary tools and the ﬁltration techniques due to Arveson to prove the connectedness

of the essential spectrum. When we consider the Discrete Schrödinger operator Hdeﬁned by

H(x(n))=(x(n−1) + x(n+1)+v(n)x(n)); x=(x(n)) ∈l2(Z),n∈Z,

with the potential sequence

(

)being periodic, there will be spectral gaps unless when

is constant

(see [4,5] for example). However, if we write the matrix representation with respect to the standard

orthonormal basis, it is tridiagonal; hence, Arveson’s techniques are available. Here the spectral analysis

depends on the nature of the potential. The spectral gap issues of such operators were studied with the

linear algebraic techniques in [6]. The spectral gap issues of arbitrary bounded self-adjoint operators

can be found in the literature (see [7,8] for example).

https://doi.org/10.17993/3ctic.2022.112.52-59

Another interesting point is to carry over such techniques to the multi-dimensional case by replacing

Zby Zn.

ACKNOWLEDGMENT

V.B. Kiran Kumar wishes to thank KSCSTE, Kerala, for ﬁnancial support via the YSA-Research

project.

REFERENCES

[1]

W.B. Arveson, (1994).

C∗

- algebras and numerical linear algebra. J. Funct. Anal. 122 , no.2, pp.

333–360.

[2]

A. Avila, and S. Jitomirskaya (2009). The Ten Martini Problem Ann. of Math. 170 , no.1, pp.

303–342.

[3]

A. Böttcher,A.V. Chithra, and M.N.N. Namboodiri (2001). Approximation of approximation

numbers by truncation Integral Equations Operator Theory 39 , pp. 387–395.

[4]

L. Golinskii,V. B. Kiran Kumar,M.N.N. Namboodiri, and S. Serra-Capizzano, (2013).

A note on a discrete version of Borg’s Theorem via Toeplitz-Laurent operators with matrix-valued

symbols Boll. Unione Mat. Ital. 39, no.1 , pp. 205–218.

[5]

V. B. Kiran Kumar,M.N.N. Namboodiri, and S. Serra-Capizzano, (2014). Perturbation of

operators and approximation of spectrum Proc. Indian Acad. Sci. Math. Sci. 124 (2014), no. 2, ,

pp. 205–224.

[6]

V. B. Kiran Kumar(2015). Truncation Method for Random Bounded Self-adjoint Operators

Banach J. Math. Anal. 9 no.3, pp. 98–113.

[7]

M.N.N. Namboodiri(2002). Truncation method for Operators with disconnected essential spec-

trum Proc.Indian Acad.Sci.(MathSci) 112 , pp. 189–193.

[8] M.N.N. Namboodiri(2005). Theory of spectral gaps- A short survey J.Analysis 12 , pp. 1–8.

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