A multi-scale analysis scheme on Abelian groups with an application to operators dual to Hill’s equation
Authors: Damanik David , Goldstein Michael , Lukic Milivoje 
Published: May 2016

We present an abstract multi-scale analysis scheme for matrix functions

(

H

ε

(
m
,
n
)

)

m
,
n
∈

T

(H_{\varepsilon }(m,n))_{m,n\in \mathfrak {T}}

, where

T

\mathfrak {T}

is an Abelian group equipped with a distance

|

⋅

|

|\cdot |

. This is an extension of the scheme developed by Damanik and Goldstein for the special case

T

=

Z

ν

\mathfrak {T} = \mathbb {Z}^\nu

.

Our main motivation for working out this extension comes from an application to matrix functions which are dual to certain Hill operators. These operators take the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper H Subscript omega overTilde Baseline y right-bracket left-parenthesis x right-parenthesis equals minus y left-parenthesis x right-parenthesis plus epsilon upper U left-parenthesis omega overTilde x right-parenthesis y left-parenthesis x right-parenthesis comma x element-of double-struck upper R comma">

[

H

ω
~

y
]
(
x
)
=
−
y
(
x
)
+
ε
U
(

ω
~

x
)
y
(
x
)
,

x
∈

R

,

[H_{\tilde \omega } y](x)= -y(x) + \varepsilon U(\tilde \omega x) y(x), \quad x \in \mathbb {R},

\]
where

U
(
θ
)

U(\theta )

is a real smooth function on the torus

T

ν

\mathbb {T}^\nu

,

ω
~

=
(

ω
~

1

,
…
,

ω
~

ν

)
∈

R

ν

\tilde \omega = (\tilde \omega _1,\dots ,\tilde \omega _\nu )\in \mathbb {R}^\nu

is a vector with rational components, and

ε
∈

R

\varepsilon \in \mathbb {R}

is a small parameter. The group in this particular case is the quotient

T

=

Z

ν

/

{
m
∈

Z

ν

:
m

ω
~

=
0
}

\mathfrak {T} = \mathbb {Z}^\nu /\{m\in \mathbb {Z}^\nu :m\tilde \omega =0\}

.

We show that the general theory indeed applies to this special case, provided that the rational frequency vector

ω
~

\tilde \omega

obeys a suitable Diophantine condition in a large box of modes. Despite the fact that in this setting the orbits

k
+
m
ω

k + m\omega

,

k
∈

R

k \in \mathbb {R}

,

m
∈

Z

ν

m \in \mathbb {Z}^\nu

are not dense, the dual eigenfunctions are exponentially localized and the eigenvalues of the operators can be described as

E
(
k
+
m
ω
)

E(k+m\omega )

, with

E
(
k
)

E(k)

being a “nice” monotonic function of the impulse

k
≥
0

k \ge 0

. This enables us to derive a description of the Floquet solutions and the band-gap structure of the spectrum, which we will use in a companion paper to develop a complete inverse spectral theory for the Sturm-Liouville equation with small quasi-periodic potential via periodic approximation of the frequency. The analysis of the gaps in the range of the function

E
(
k
)

E(k)

plays a crucial role in this approach.

Although we are mostly interested in the case of analytic

U
U

, we need to analyze, for technical reasons, the functions

U
U

with sub-exponentially decaying Fourier coefficients in the current work.

The main novelty for the experts in this field is that, while all known multi-scale schemes run for an irrational Diophantine frequencies vector, we can run the analysis for a rational vector. The main reason why we are able to do this is that the abstract scheme in the method by Damanik and Goldstein (2014) is flexible enough to accommodate this case. In the current work we verify that one can put the case in question in the setup of the above-mentioned work.

The verification carried out in the current work also gives a strong indication that the method of Damanik and Goldstein can be applied to limit-periodic potentials.