Inner and outer continuity of the spectra for families of magnetic operators on $\mathbb{Z}^{d}$ (Spectral and Scattering Theory and Related Topics)

Inner

and

outer

_continuity

of

the

_spectra

for families of

_magnetic

_operators

on

\mathbb{Z}^{d}

S. Richard

*

Graduate school

_of

mathematics,

Nagoya University, Chikusa‐ku,

Nagoya

464‐S602,

Japan;

On leave

_of

absence

_from

Univ.

_Lyon,

Uni‐

versité Claude Bernard

Lyon 1,

CNRS UMR

_5208,

Institut Camille

Jordan, 43

blvd. du11 novembre 1918, F‐69622 Villeurbanne

cedex,

France.

E‐‐mails:

_{richard@math.nagoya‐u.ac.}

jp

Abstract

In thisnoteweconsider_{magnetic self‐adjointoperators}on\mathbb{Z}^{d}whosesymbols

and_magneticfields _dependon a_parameter $\epsilon$. Sufficient conditions areimposed

such that the_spectrumof these operatorsvaries _continuouslywith_respect to $\epsilon$.

The _emphasizeis _put on a constructionwhich is independent of_any particular

choice of the_{magneticpotentials.}

2010 Mathematics

_Subject

Classification:

_{81\mathrm{Q}10,}

47\mathrm{L}65

Keywords:

Discrete_operators,

_magnetic

_field,

_spectrum, twistedcrossed

_product

algebra

1

Introduction

This paper is an extended version of a

presentation

made at the conference

_Spectral

and

_{Scattering Theory}

and Related

Topics

at Rims in

_Kyoto

in

_January

2016. The

presentation

wasbasedon the_paper

[11]

towhichwerefer for moredetails andfor the

proofs.

In the Hilbert _space \mathcal{H}

_{:=l^{2}(\mathbb{Z}^{d})}

and for some fixed parameter $\epsilon$ let us consider

operatorsof the form

[H^{ $\epsilon$}u](x):=\displaystyle \sum_{y\in \mathbb{Z}^{d}}h^{ $\epsilon$}(x;y-x)\mathrm{e}^{i$\phi$^{ $\epsilon$}(x,y)}u(y)

(1.1)

with u\in \mathcal{H} of finitesupport,

x\in \mathbb{Z}^{d}

and where h^{ $\epsilon$} :

\mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow \mathbb{C}

and

$\phi$^{ $\epsilon$}

:

\mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow \mathbb{R}

satisfy

*

(i)

\displaystyle \sum_{x\in \mathbb{Z}^{d}}\sup_{q\in \mathrm{Z}^{d}}|h^{ $\epsilon$}(q;x)|<\infty,

(ii) h^{ $\epsilon$}(q+x;-x)=h^{ $\epsilon$}(q;x)

for any q,

x\in \mathbb{Z}^{d},

(iii) $\phi$^{ $\epsilon$}(x, y)=-$\phi$^{ $\epsilon$}(y, x)

for allx,

y\in \mathbb{Z}^{d}.

Such _operators are

usually

called discrete

magnetic

Schrödinger

operators. Note that

condition

_(i)

ensures that H^{ $\epsilon$} extends

continuously

to a bounded _operator in \mathcal{H}, and

can be

simply

rewritten as

h^{ $\epsilon$}\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))

. Conditions

(ii)

and

(iii)

imply

that

the _operator H^{ $\epsilon$} is

_{self‐adjoint.}

Note also that a _map

$\phi$

: \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow \mathbb{R}

satisfying

$\phi$(x, y)=- $\phi$(y, x)

foranyx,

y\in \mathbb{Z}^{d}

will be called a

magnetic

potential.

Our aim is exhibit some

continuity properties

of the _spectrum of these _operators

under suitable and natural

_assumptions.

Natural conditionsonthe

family

of

symbols

h^{ $\epsilon$}

are

imposed

below.

However,

it is well‐known

(at

least inthe continuous

setting)

that

continuity

conditions should not be

_imposed

on the

magnetic

potentials

but rather on

the

_magnetic

fields. This

_requirement

comes from the

non‐unicity

for the choice ofa

magnetic

potential. Indeed,

if

_$\phi$

: \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow \mathbb{R} isdefined for_anyx,

y\in \mathbb{Z}^{d}

by

$\phi$'(x, y)\equiv[ $\phi$+\nabla $\varphi$](x, y) := $\phi$(x, y)+ $\varphi$(y)- $\varphi$(x)

(1.2)

forsome _$\varphi$: \mathbb{Z}^{d}\rightarrow \mathbb{R}_,then the

magnetic

_operatorsconstructed with

$\phi$

and

$\phi$'

areknown

to be

_unitarily

_equivalent.

This_propertyiscalled the gauge invariance of the

magnetic

operatorsand

_imposes

a

slightly

moreelaborated notionof

continuity

for the

magnetic

contribution,

as

emphasized

below.

Before

_explaining

moreindetails the_necessaryconstruction,letusstatea

simplified

version of our main theorem in which the $\epsilon$

‐dependence

on

$\phi$^{ $\epsilon$}

is _very

simple.

A more

general

setting

will be introduced in the

_subsequent

sections. The

_continuity

we shall

consider for the_spectrum

_corresponds

tothe

_stability

of the

_spectral

gapsaswell asthe

stability

of the

_spectral

_compounds.

Ina more

_precise

terminology

weshall proveinner

andouter

_continuity

for the

_family

of_spectra.The

_following

definitionisborrowed from

[1]

but

_{originally inspired by}

_[3].

Definition 1.1. Let $\Omega$ be a _compact

Hausdorff

_space, and let

\{$\sigma$_{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$}

be a

family of

closed subsets

_of

\mathbb{R}.

1. The

_family

_{\{$\sigma$_{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$}}

is outer continuous at

_{$\epsilon$_{0}\in $\Omega$}

_{if for}

_anycompact subset \mathcal{K}

_of

\mathbb{R} such that

_{\mathcal{K}\cap$\sigma$_{ $\epsilon$ 0}=\emptyset}

there exists a

neighbourhood

\mathcal{N}=\mathcal{N}(\mathcal{K}, $\epsilon$_{0})

of

_{$\epsilon$_{0}} in $\Omega$

such that

_{\mathcal{K}\cap$\sigma$_{ $\epsilon$}=\emptyset for}

_any

_{$\epsilon$\in \mathcal{N},}

2. The

_family

_{\{$\sigma$_{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$}}

is inner continuous at

_{$\epsilon$_{0}\in $\Omega$}

_{if for}

_{any open} subset \mathcal{O}

_of

\mathbb{R}

such that

_{\mathcal{O}\cap$\sigma$_{ $\epsilon$ 0}\neq\emptyset}

there exists a

neighbourhood

\mathcal{N}=\mathcal{N}(\mathcal{O}, $\epsilon$_{0})

of

_{$\epsilon$_{0}} in $\Omega$ such

that

_{\mathcal{O}\cap$\sigma$_{ $\epsilon$}\neq\emptyset}

_for

_any $\epsilon$\in \mathcal{N}.

With this definition at hand we can now choose $\Omega$

:=[0

,1

]

and state a

special

instanceofourmainresult whichis

presented

inTheorem3.1. Note that the

_following

statement is

_inspired

from

_[8]

and a

comparison

with the

existing

literature will be

Theorem 1.2. For each $\epsilon$\in $\Omega$

_:=[0

,1

]

let h^{ $\epsilon$} :

\mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow \mathbb{C}

satisfy

the aboveconditions

(i)

and

_(ii),

and

1)

_{\displaystyle \sum_{x\in \mathrm{Z}^{d}}\sup_{q\in \mathbb{Z}^{d}}\sup_{ $\epsilon$\in[0,1]}|h^{ $\epsilon$}(q;x)|<\infty,}

2)

Forany

fixed

x\in \mathbb{Z}^{d},

\displaystyle \lim_{ $\epsilon$\rightarrow $\epsilon$}\sup_{q\in \mathbb{Z}^{d}}|h^{$\epsilon$'}(q;x)-h^{ $\epsilon$}(q;x)|=0.

Let also

_$\phi$

be a

_magnetic

potential

which

satisfies

| $\phi$(x, y)+ $\phi$(y, z)+ $\phi$(z, x)|\leq

area

\triangle(x, y, z)

,

where

_{\triangle(x, y, z)}

means the

triangle

in\mathbb{R}^{d} determined

by

the three

_points

_{x, y,}

z\in \mathbb{Z}^{d}.

Then

_for

H^{ $\epsilon$}

_defined

onu\in \mathcal{H}

by

[H^{ $\epsilon$}u](x):=\displaystyle \sum_{y\in \mathrm{Z}^{d}}h^{ $\epsilon$}(x;y-x)\mathrm{e}^{i $\epsilon \phi$(x,y)}u(y)

the

_{family of}

_spectra

_{$\sigma$(H^{ $\epsilon$})}

_forms

an outer and an inner continuous

family

at every

points

$\epsilon$\in $\Omega$.

It is

_{certainly impossible}

to mention all _papers

_dealing

with

_{continuity properties}

of families of such _operators, but let us cite a few of them which are relevant for our

investigations.

First of

_all,

let us mention the seminal _paper

[3]

in which the author

proves the

Lipschitz continuity

of gap boundaries with respect to the variation of a

constant

_magnetic

field fora

family

of

pseudodifferential

_{operators acting}

on \mathbb{Z}^{2}. In

[6]

and basedontheframework introducedin

[13],

similar

Lipschitz

_continuity

is

proved

for

self‐adjoint

operators

acting

on a

crystal lattice,

anatural

generalization

of

\mathbb{Z}^{d}

.

Papers

[8]

and

_[5]

also deal with families of

_magnetic

_{pseudodifferential}

_operatorson\mathbb{Z}^{2}, and the

results containedin

_[8]

_partially

motivatedourwork. Let usstillmentiontwoadditional

paperswhichareattherootofourwork:

[7]

inwhicha

general

framework for

magnetic

systems,

involving

twisted crossed

_product

C^{*}

_‐algebras,

is

_introduced,

and the reference

[1]

which containsresults similar toours but in acontinuous

setting.

Let us now

emphasize

that the framework

_presented

in Section 3 does not allow

usto_{get any}

quantitative estimate,

as

emphasized

in the_{recent paper}

[2].

Indeed,

the

veryweak

continuity requirement

we

impose

onthe $\epsilon$

‐dependence

onour

objets

cannot

lead to any

Lipschitz

or Hölder

continuity.

More

_{stringent assumptions}

are _necessary

for that purpose, and suchestimates

certainly

deserve further

investigations.

Our

_approach

relies on the _concepts of twisted crossed

product

C^{*}

‐algebras

and

on a field of such

algebras, mainly

borrowed from

[12,

14].

In the discrete

_setting,

such

_algebras

have

_already

been

_used,

for

_example

in

_[3,

_6,

_13].

_However,

instead of

considering

a

2‐cocycle

with scalar

values,

which issufficient for thecase ofaconstant

This allowsus toconsider

_arbitrary

magnetic

potential

on \mathbb{Z}^{d}and toencompass all the

corresponding

operatorsin a

single algebra.

Letus

finally

describe thecontentof thispaper.In Section2weintroduce theframe‐

work fora

single

_magnetic

_system, i.e.for afixed $\epsilon$.Forthat reason,no $\epsilon$

‐dependence

is

indicatedinthissection. InSection3the- $\epsilon$

‐dependence

isintroduced and thecontinuous

dependence

onthis_parameterisstudied. Ourmainresult is

presented

in Theorem 3.1.

2

Discrete

_magnetic

_systems

In thissection weintroduce a

quantity

which does not

depend

on the choice ofa _par‐

ticular

_magnetic

_potential.

In that_respect,thisfunction

_{depends only}

onthe

magnetic

field,

as

emphasized

below. Note that no $\epsilon$

‐dependence

is written in thissection.

We start

_{by recalling}

that a

magnetic

potential

consists ina_map

$\phi$

:

\mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow \mathbb{R}

satisfying

for _{any x,}

_{y\in \mathbb{Z}^{d}}

the relation

$\phi$(x, y)=- $\phi$(y, x)

.

Then,

given

such a

magnetic

potential $\phi$

let usintroduce and

study

anew_map

$\omega$:\mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow l^{\infty}(\mathbb{Z}^{d})

defined forq,x,

y\in \mathbb{Z}^{d}

by

[ $\omega$(x, y)](q)\equiv $\omega$(q;x, y)

:=\exp\{i[ $\phi$(q, q+x)+ $\phi$(q+x, q+x+y)+ $\phi$(q+x+y, q)]\}.

(2.1)

Note that

_{$\omega$(x, y)}

is

_{unitary‐valued}

since

_{| $\omega$(q, ; x, y)|=1}

forany

q\in \mathbb{Z}^{d}.

\ln

fact,

$\omega$(x, y)

takes values in the

_unitary

group of the

algebra

l^{\infty}(\mathbb{Z}^{d})

, which shall

simply

denote

by

\mathscr{U}(\mathbb{Z}^{d})

, i.e.

%(Zd) =\{f:\mathbb{Z}^{d}\rightarrow \mathrm{T}\}.

Let usalso introduce theaction $\theta$ of\mathbb{Z}^{d}

by translations, namely

on_any

f\in l^{\infty}(\mathbb{Z}^{d})

one sets

$\theta$_{x}f(y)=f(x+y)

.

In

_particular,

since

_{$\omega$(x, y)\in l^{\infty}(\mathbb{Z}^{d})}

we have

[$\theta$_{z} $\omega$(x, y)](q):=[ $\omega$(x, y)](q+z)= $\omega$(q+z;x, y)

.

Basedon these

definitions,

the

following

_properties

for $\omega$ canbe obtained

by straight‐

forward

_{computations.}

Recall that the notation

_{$\phi$+\nabla $\varphi$}

has been introduced in

_(1.2).

Lemma 2.1. Let

_$\phi$

be a

magnetic

potential

and let $\omega$ be

defined by

(2.1).

Then

for

_any

x, y,

z\in \mathbb{Z}^{d}

the

following

properties hold:

(i)

$\omega$(x+y, z) $\omega$(x, y)=$\theta$_{x} $\omega$(y, z) $\omega$(x, y+z)

(2‐cocycle property)

(iii)

$\omega$(x, -x)=1

(additional property)

(iv)

For _{any $\varphi$} :

\mathbb{Z}^{d}\rightarrow \mathbb{R}

, the

magnetic

potentials

$\phi$

and

$\phi$+\nabla $\varphi$ define

the same

function

$\omega$

(independence property).

With the above

_lemma,

one

directly

infers that $\omega$ is anormalized

2‐cocycle

on \mathbb{Z}^{d}

with valuesin

_{\mathscr{U}(\mathbb{Z}^{d})}

and which satisfies the additional_property

_(iii).

In

_addition,

this

map

depends only

on

equivalent

classes of

magnetic

potentials,

as

emphasized

in

(iv).

One couldarguethat the

2‐cocycle

$\omega$

depends only

onthe

_magnetic

field

asintroduced

in

_[4],

and not on the choice ofa

magnetic

potential.

However,

this would lead us too

far fromour_{purpose since}wewould havetoconsider\mathbb{Z}^{d}as a

graph

endowedwith

edges

betweenevery

pair

ofvertices.

Let us now

adopt

a _very

pragmatic point

ofviewand recall

only

the

strictly

nec‐

essaryinformation on twisted crossed

product

C^{*}

‐algebras.

More can be found in the

fundamental papers

[9, 10]

or in the _{review paper}

[7].

Since the _group we are

dealing

with is

_simply

\mathbb{Z}^{d}

, most of the necessary informationcan also Ue foundin

[14].

Consider

_{l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))}

, with the norm

\displaystyle \Vert f\Vert_{1,\infty}:=\sum_{x\in \mathrm{Z}^{d}}\sup_{q\in \mathrm{Z}^{d}}|f(q;x)| \forall f\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))

,

and with the twisted

_product

and the involution defined for _any

_f,

_{g\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))}

by

[f\displaystyle \mathrm{o}g](q;x):=\sum_{y\in \mathrm{Z}^{d}}f(q;y)g(q+y;x-y) $\omega$(q;y, x-y)

and

f^{ $\theta$}(q;x)=\overline{f(q+x;-x)}.

Endowed with this

_{multiplication}

and with this

_involution,

_{l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))}

is aBanach

*

‐algebra.

The

_{corresponding enveloping}

_C^{*}

_‐algebra

_{will be denoted}

_by

_{\mathbb{C}( $\omega$)}

. Recall

that this

_{algebra corresponds}

to the

_completion

of

_{l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))}

with _respect to the

C’‐norm definedasthe_supremumoverall the faithful

representations

of

l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))

.

Asa_consequence,

l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))

is dense in

_{\not\subset( $\omega$)}

and the new C^{*}‐norm

\Vert\cdot\Vert

satisfies

\Vert f\Vert\leq\Vert f\Vert_{1,\infty}.

Note that the above construction holds for _any normalized

_{2‐cocycle satisfying}

the additional _property

_(iii).

In

_fact,

it is

_proved

in

_[11]

that any such

2‐cocycle

can

be obtained

_by

a

magnetic

potential,

i.e. there

_always

exists a

_magnetic

potential $\phi$

satisfying

(2.1).

Letus nowlookatafaithful_{representation} of the

_algebra

\mathbb{C}( $\omega$)

inthe Hilbertspace

\mathcal{H}=l^{2}(\mathbb{Z}^{d})

. For that purpose, consider any

magnetic potential

$\phi$ satisfying

\exp\{i[ $\phi$(q, q+x)+$\phi$'(q+x, q+x+y)+$\phi$'(q+x+y, q)]\}= $\omega$(q;x, y)

.

(2.2)

Then for_any

_{h\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))}

, anyu\in \mathcal{H} and any

x\in \mathbb{Z}^{d}

onesets

Clearly,

this

_expression

is similar to theone contained in

(1.1)

and this fact

partially

justifies

theentire construction. Themain

_properties

of this

_{representation}

are

gathered

inthe

_following

statement, which

corresponds

to

_[7,

_Prop.

2. 16 &

_2.17]

_adapted

to our

setting.

Proposition

2.2. Let

_$\phi$

be any

magnetic

potential

satisfying

(2.2)

for

a

given

normal‐

ized

_2‐cocycle

$\omega$ with the additional condition

(iii)

of

Lemma 2.1.

Then,

(i)

The

_{representation}

\Re \mathrm{e}\mathfrak{p}^{$\phi$'}

_of

_{\mathbb{C}( $\omega$)}

isirreducible and

_faithful

(ii)

Any

other choice

_for

_$\phi$'

leadsto a

unitarily equivalent

representation,

(iii)

If

_{h\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))}

, then

\Re e\mathrm{p}^{$\phi$'}(h)\dot{u}

self‐adjoint if

h^{ $\theta$}=h.

3

A continuous field of C^{*}

_‐algebras

In this section we consider a

family

of discrete

magnetic

systems which are _parame‐

terized

_by

the elements $\epsilon$ of a _compact Hausdorff _space $\Omega$. The necessary

continuity

relations between thevarious

_objects

is encoded in the structure of a field of twisted

crossed

_product

C^{*}

_‐algebras,

asintroduced in

[12]

and

already

used in a similar con‐

text in

_[1].

_Again,

let us be _very

pragmatic

and introduce

only

the

strictly

necessary

information,

and refer to

_[11,

Sec.

_3]

formoredetails.

For fixed x,

y\in \mathbb{Z}^{d}

, let usconsider acontinuous map

$\Omega$\ni $\epsilon$\mapsto$\omega$^{ $\epsilon$}(x, y)\in \mathscr{U}(\mathbb{Z}^{d})

(3.1)

such that each element$\omega$^{ $\epsilon$}

₎

isa

2‐cocycle

on\mathbb{Z}^{d}with values %

(\mathbb{Z}^{d})

andwhich satisfies

the.additional

property

(iii)

of Lemma2.1. Note that

_equivalently

one can consider a

normalized

_2‐cocycle

$\omega$ on \mathbb{Z}^{d} and with valuesin theC^{*}

‐algebra

C( $\Omega$;l^{\infty}(\mathbb{Z}^{d}))

. In this

setting

the additional condition reads

_{$\omega$(x, -x)=1}

for any

x\in \mathbb{Z}^{d}

, and the

following

relation

_clearly

holds:

$\omega$^{ $\epsilon$}(q;x, y)\equiv[$\omega$^{ $\epsilon$}(x, y)](q)=[ $\omega$(x, y)]( $\epsilon$, q)\equiv $\omega$( $\epsilon$, q;x, y)

.

Let usstressthat the

_continuity

_assumption

mentioned inthe

_previous

_paragraph

is

_precisely

the sufficient one for the

continuity

of the _spectrum. More

precisely,

this

continuity

condition

_corresponds

to a

gauge‐independent

condition,

and does not cor‐

respond

to any direct

_requirement

on _any

magnetic

potential $\phi$^{ $\epsilon$}

.

So,

let us now state

the mainresult of this note:

Theorem3.1. Let

_{\{$\omega$^{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$}}

bea

family of

normalized

2‐cocycle

satisfying

the additional

property

(iii)

of

Lemma2.1 and such that the map

defined by

(3.1)

is continuous. Con‐

sidera

family

\{h^{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$}\subset l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}))

such that the

following

conditions are

satisfied:

(ii)

For _any

_fixed

_{x\in \mathbb{Z}^{d},}

\displaystyle \lim_{ $\epsilon$\rightarrow $\epsilon$}\sup_{q\in \mathbb{Z}^{d}}|h^{$\epsilon$'}(q;x)-h^{ $\epsilon$}(q;x)|=0,

(iii) (h^{ $\epsilon$})^{ $\theta$}=h^{ $\epsilon$}.

Then, for

any

family of

magnetic

potential $\phi$^{ $\epsilon$}

satisfying

[$\omega$^{ $\epsilon$}(x, y)](q)=\exp\{i[$\phi$^{ $\epsilon$}(q, q+x)+$\phi$^{ $\epsilon$}(q+x, q+x+y)+$\phi$^{ $\epsilon$}(q+x+y, q)]\},

the

_{family of}

_spectra

\{ $\sigma$(\Re \mathrm{t}\mathfrak{p}^{$\phi$^{ $\epsilon$}}(h^{ $\epsilon$}))\}_{ $\epsilon$\in $\Omega$}

_forms

anouterandaninner continuous

family

at every

point

$\epsilon$

of

$\Omega$.

The

_proof

of this statement as well as the

proof

of Theorem 1.2 are

given

in

[11].

Let us

simply

mentionthatit isbasedonthe constructionof the

enveloping

C^{*}

‐algebra

\mathbb{C}( $\omega$)

of the Banach *

‐algebra

l^{1}(\mathbb{Z}^{d};C( $\Omega$;l^{\infty}(\mathbb{Z}^{d})))

endowed with a

product

and an

involution defined in termsof $\omega$.

Properties

of the evaluation maps e_{ $\epsilon$} :

\mathrm{C}( $\omega$)\rightarrow \mathrm{C}($\omega$^{ $\epsilon$})

at any $\epsilon$\in $\Omega$ lead rather

_{straightforwardly}

to our main

theorem,

once the abstract

results obtained in

_[12]

aretakeninto account.

References

[1]

N.

_Athmouni,

M.

_Măntoiu,

R.

_Purice,

On the

_continuity

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_spectra

_{for families}

_of

magnetic

pseudodifferential

operators, J. Math.

_Phys.

51 no. 8

(2010),

083517,

15

pp.

[2]

S.

_Beckus,

J.

_{Bellissard, Continuity of}

the _spectrum

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a

field of self‐ajdoint

_opera‐

tors,

Preprint

ArXiv 1507.04641.

[3]

J.

_{Bellissard, Lipshitz}

_continuity

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