On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters
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irk-123456789-1066452016-10-02T03:02:30Z On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters Golinskii, L. Kheifets, A. Peherstorfer, F. Yuditskii, P. 2010 Article On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters / L. Golinskii, A. Kheifets, F. Peherstorfer, P. Yuditskii // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 277-290. — Бібліогр.: 12 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106645 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Golinskii, L. Kheifets, A. Peherstorfer, F. Yuditskii, P. On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters Журнал математической физики, анализа, геометрии |
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Golinskii, L. Kheifets, A. Peherstorfer, F. Yuditskii, P. |
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On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters |
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On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters |
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On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters |
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On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters |
| title_full_unstemmed |
On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters |
| title_sort |
on a class of verblunsky parameters that corresponds to guseinov's class of jacobi parameters |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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| citation_txt |
On a Class of Verblunsky Parameters that Corresponds to Guseinov's Class of Jacobi Parameters / L. Golinskii, A. Kheifets, F. Peherstorfer, P. Yuditskii // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 277-290. — Бібліогр.: 12 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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Journal of Mathematical Physics, Analysis, Geometry
2010, v. 6, No. 3, pp. 277–290
On a Class of Verblunsky Parameters that Corresponds
to Guseinov’s Class of Jacobi Parameters
L. Golinskii
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:golinsky@ilt.kharkov.ua
A. Kheifets∗
Department of Mathematics
University of Massachusetts Lowell, 01854, USA
E-mail:Alexander Kheifets@uml.edu
F. Peherstorfer ∗∗ and P. Yuditskii∗∗
Institute for Analysis, Johannes Kepler University Linz
A-4040 Linz, Austria
E-mail:franz.peherstorfer@jk.uni-linz.ac.at
petro.yuditskiy@jku.at
Received August 7, 2009
The direct and inverse Geronimus relations between Verblunsky parame-
ters of measures on the unit circle and Jacobi parameters of their Szegő trans-
forms have been used to prove that Guseinov’s class of Jacobi parameters
∞∑
n=0
n(|an−1|+|bn|) < ∞ is in a canonical correspondence with the following
class of Verblunsky parameters αn → 0 and
∞∑
n=0
n|αn+2 − αn| < ∞.
Key words: Verblunsky coefficients, Szegő transform, direct and inverse
Geronimus relations, Jacobi matrices; spectral measure.
Mathematics Subject Classification 2000: 47B36 (primary); 42C05
(secondary).
∗The work was partially supported by the University of Massachusetts Lowell Research and
Scholarship Grant, project number: H50090000000010.
∗∗The work was partially supported by the Austrian Science Found FWF, project number:
P20413–N18.
c© L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii, 2010
L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii
1. Introduction
In the late 1950s L. Faddeev [1] developed a scattering theory for the one-
dimensional Schrödinder equation
−y′′ + q(x)y = λ2y (1.1)
under the following assumption on the potential q
∞∫
−∞
(1 + |x|)|q(x)|dx < ∞. (1.2)
In 1979 Deift and Trubowitz [2] found a gap in Faddeev’s construction and
analyzed completely the case of potentials with a finite second moment
∞∫
−∞
(1 + x2)|q(x)|dx < ∞. (1.3)
As it turned out, they were unaware of the book [3] by V.A. Marchenko where
he had given a correct proof of Faddeev’s theorem for the class of potentials (1.2).
That is why we call (1.2) the Faddeev–Marchenko condition.
In the mid 1970s Guseinov [4, 5] suggested a discrete version of the Faddeev–
Marchenko theory for Jacobi matrices
an−1yn−1 + bnyn + anyn+1 = (λ + 1/λ)yn (1.4)
under the condition ∞∑
n=0
n(|an − 1|+ |bn|) < ∞. (1.5)
We say that a Jacobi matrix J = J({an}, {bn}) belongs to Guseinov’s class (G)
if its parameters satisfy (1.5).
In his recent papers [6–8] and PhD thesis [9] E. Ryckman came up with a new
class of Jacobi matrices for which a complete spectral description is available.
Moreover, this class extends Guseinov’s class (see Prop. 3.3 below).
Let us write
β = {βn} ∈ `2
s if ‖β‖2
`2s
:=
∑
n
|n|s|βn|2 < ∞.
Definition 1.1. A Jacobi matrix is said to be in Ryckman’s class (R), or
a, b ∈ (R), if the series
∑
n(a2
n − 1) and
∑
n bn are conditionally summable, and
λn := −
∞∑
k=n+1
bk ∈ `2
1, κn := −
∞∑
k=n+1
(a2
k − 1) ∈ `2
1.
278 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
On a Class of Verblunsky Parameters
Ryckman’s argument is based on two main ingredients. First, he shows that
under an appropriate Szegő transform the spectral measures of the class (R)
correspond to measures µ on the unit circle that are from the B. Golinskii–
Ibragimov (GI) class:
∞∑
n=1
n|αn|2 < ∞,
where αn are the Verblunsky parameters of µ. The second step in Ryckman’s
argument is the Strong Szegő theorem, which provides a complete spectral cha-
racteristic of (GI) class.
A problem we address in this note is to find the class of measures on the
unit circle that corresponds to Guseinov’s class (1.5) under an appropriate Szegő
transform. It turns out that this class is described as follows.
Definition 1.2. We say that µ ∈ (K) (or α = {αn} ∈ (K)) if
αn → 0 and
∞∑
n=0
n|αn+2 − αn| < ∞. (1.6)
(K) is a proper subclass of (GI) class, and it solves the above problem.
Ryckman also studies the class of Jacobi matrices with {λn}, {κn} ∈ `2
1 ∩ `1
and shows that the corresponding class of measures on the unit circle satisfies
{αn} ∈ `2
1 ∩ `1. It is an open problem to characterize a class of Jacobi matrices
which corresponds in this sense to the whole Baxter’s class {αn} ∈ `1.
Note that the scattering theory for orthogonal polynomials on the unit circle
(CMV matrices in the modern terminology) and for Jacobi matrices was suggested
by Geronimo and Case in [10] for Baxter’s class, and in [11], for Guseinov’s class.
2. Classes of Verblunsky Parameters
Theorem 2.1 (Szegő). Let µ be a nontrivial probability measure on T with
Verblunsky parameters {αn}, |αn| < 1. Then
∞∑
n=0
|αn|2 < ∞
if and only if µ(dt) = w(t)m(dt), m(dt) the normalized Lebesgue measure on T,
with log w ∈ L1.
Theorem 2.2 (B. Golinskii–Ibragimov’s version of the Strong Szegő theo-
rem). Let µ be a nontrivial probability measure on T with the Verblunsky para-
meters {αn}, |αn| < 1. The following assertions are equivalent:
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 279
L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii
1. α ∈ `2
1, i.e.,
∞∑
n=0
n|αn|2 < ∞.
2. µ(dt) = w(t)m(dt) with l̂og w ∈ `2
1. Here f̂ is a sequence of Fourier coeffi-
cients of a function f .
Proposition 2.3. α ∈ (K) =⇒ α ∈ `2
1.
P r o o f. The claim will be proved separately for even and odd n’s. So we
want to show that
cn → 0 and
∞∑
n=1
n|cn+1 − cn| < ∞ =⇒
∞∑
n=1
n|cn|2 < ∞.
We have that
∞∑
n=1
n|cn|2 =
∞∑
n=1
n|
∞∑
k=n
(ck+1 − ck)|2 ≤
∞∑
n=1
n(
∞∑
k=n
|ck+1 − ck|)2
=
∞∑
n=1
n(
∞∑
k=n
|ck+1 − ck|)(
∞∑
l=n
|cl+1 − cl|) =
∞∑
n=1
∞∑
k=n
∞∑
l=n
n|ck+1 − ck||cl+1 − cl|
≤
∞∑
n=1
∞∑
k=n
∞∑
l=n
l|ck+1 − ck||cl+1 − cl| ≤
∞∑
n=1
∞∑
k=n
∞∑
l=1
l|ck+1 − ck||cl+1 − cl|
=
∞∑
k=1
k∑
n=1
∞∑
l=1
l|ck+1 − ck||cl+1 − cl| =
∞∑
k=1
k
∞∑
l=1
l|ck+1 − ck||cl+1 − cl|
=(
∞∑
k=1
k|ck+1 − ck|)2.
It is easy to see that also α ∈ (K) =⇒ α ∈ `1. Indeed,
|αn| = |
∞∑
k=n
(αk − αk+2)|,
and so ∞∑
n=0
|αn| ≤
∞∑
n=0
n|αn − αn+2|.
Hence (K) ⊂ `2
1 ∩ `1. Obviously, the inclusion is proper.
280 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
On a Class of Verblunsky Parameters
3. Classes of Jacobi Parameters
Definition 3.1. We say that a, b ∈ (S) if
∞∑
n=0
(|an − 1|2 + |bn|2) < ∞.
We say that a, b ∈ (R) if
∞∑
n=0
n
∣∣∣∣∣
∞∑
k=n
(ak − 1)
∣∣∣∣∣
2
< ∞,
∞∑
n=0
n
∣∣∣∣∣
∞∑
k=n
bk
∣∣∣∣∣
2
< ∞.
We say that a, b ∈ (G) if
∞∑
n=0
n(|an − 1|+ |bn|) < ∞.
Proposition 3.2. In Definition 3.1 one can everywhere equivalently replace
an − 1 with a2
n − 1.
P r o o f. It is obvious for (S) and (G), we prove it for (R). Assume that
∞∑
n=0
n
∣∣∣∣∣
∞∑
k=n
(ak − 1)
∣∣∣∣∣
2
< ∞.
Define cn as
cn =
∞∑
k=n
(ak − 1). (3.1)
By assumption cn ∈ `2
1. Then an − 1 = cn − cn+1 ∈ `2
1. Define dn as
dn =
∞∑
k=n
(ak − 1)2.
Then, by Lemma 2.4 of [7], dn ∈ `2
1. Therefore,
fn =
∞∑
k=n
(a2
k − 1) = dn + 2cn ∈ `2
1.
In other words,
∞∑
n=0
n
∣∣∣∣∣
∞∑
k=n
(a2
k − 1)
∣∣∣∣∣
2
< ∞.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 281
L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii
Conversely, assume that
∞∑
n=0
n|
∞∑
k=n
(a2
k − 1)|2 < ∞.
Define fn as
fn =
∞∑
k=n
(a2
k − 1).
By assumption fn ∈ `2
1. Then a2
n − 1 = fn − fn+1 ∈ `2
1. By Lemma 2.4 of [7],
∞∑
k=n
(a2
k − 1)2 ∈ `2
1.
Since
dn :=
∞∑
k=n
(ak − 1)2 <
∞∑
k=n
(ak − 1)2(ak + 1)2 =
∞∑
k=n
(a2
k − 1)2 ∈ `2
1,
we get that dn ∈ `2
1. Therefore,
cn :=
∞∑
k=n
(ak − 1) =
1
2
(fn − dn) ∈ `2
1
In other words,
∞∑
n=0
n
∣∣∣∣∣
∞∑
k=n
(ak − 1)
∣∣∣∣∣
2
< ∞.
Proposition 3.3. a, b ∈ (G) =⇒ a, b ∈ (R) =⇒ a, b ∈ (S).
P r o o f. The proof of the first implication is similar to the proof of
Proposition 2.3 above. To prove the second implication take cn (3.1). By (R)
cn ∈ `2
1; therefore,
an − 1 = cn − cn+1 ∈ `2
1.
Then an − 1 ∈ `2, which means (S). The proof for bn is similar.
282 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
On a Class of Verblunsky Parameters
4. Direct Geronimus Relations
We assume that
w(t̄) = w(t), |t| = 1,
or equivalently, all Verblunsky parameters αn are real numbers, and we assume
that
w(t)m(dt) = w(t)
dt
2πit
is a probability measure on T. The image measure (on [−2, 2]) of this measure
under the mapping x = t + 1
t is
w(t(x))
dx
π
√
4− x2
. (4.1)
Therefore, it is also a probability measure on [−2, 2].
For γ1, γ2 = ±1 we define four measures on [−2, 2]
ρ(x)dx =c(γ1, γ2, w) w(t(x))
√
(2− x)γ1
√
(2 + x)γ2 dx
=c(γ1, γ2, w) w(t(x))
√
(2− x)γ1+1
√
(2 + x)γ2+1
dx√
4− x2
,
where the constants c(γ1, γ2, w) are chosen such that the measures are probability
measures. The later is possible since the measures are finite. For instance, if
γ1 = γ2 = −1 and c = 1
π , we get the above image measure (4.1). These four
transforms are called the Szegő transforms.
The following equalities are known as the direct Geronimus relations (cf., [12,
Ths. 13.1.7 and 13.2.1]).
Proposition 4.1 (Direct Geronimus Relations). Jacobi parameters an and bn
of the measure ρ(x)dx are expressed in terms of the Verblunsky parameters of the
measure w(t)m(dt) as follows:
In the case γ1 = γ2 = −1 they are
[a(e)
n+1]
2 =(1− α2n−1)(1− α2
2n)(1 + α2n+1),
b
(e)
n+1 =(1− α2n−1)α2n − (1 + α2n−1)α2n−2.
In the case γ1 = γ2 = 1 they are
[a(o)
n+1]
2 =(1 + α2n+1)(1− α2
2n+2)(1− α2n+3),
b
(o)
n+1 =− (1 + α2n+1)α2n+2 + (1− α2n+1)α2n.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 283
L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii
In the case γ1 = −γ2 = ±1 they are
[a(±)
n+1]
2 =(1± α2n)(1− α2
2n+1)(1∓ α2n+2),
b
(±)
n+1 =∓ (1± α2n)α2n+1 −±(1∓ α2n)α2n−1.
Here n = 0, 1, . . ., α−1 = −1.
Proposition 4.2.
α ∈ `2
1 =⇒ a, b ∈ (R),
α ∈ (K) =⇒ a, b ∈ (G).
P r o o f. We consider case (e), other ones are similar. By Proposition 4.1
[a(e)
n+1]
2 =(1− α2n−1)(1− α2
2n)(1 + α2n+1) = 1− α2n−1 + α2n+1
−α2
2n − α2n−1α2n+1 + α2n−1α
2
2n − α2n+1α
2
2n + α2n−1α
2
2nα2n+1.
Let αn ∈ `2
1. Consider
∞∑
k=n
(a2
k+1 − 1) = −α2n−1 +
∞∑
k=n
. . .
All terms in the sum on the right are either ”quadratic” in α or dominated by
terms ”quadratic” in α. By Lemma 2.4 of [7], the sequence on the right is in `2
1.
Therefore,
∞∑
k=n
(a2
k+1 − 1) ∈ `2
1,
meaning that an ∈ (R).
Let αn ∈ (K), then
|a2
n+1 − 1| ≤| − α2n−1 + α2n+1|
+|α2n|2 + |α2n−1α2n+1|+ |α2n−1 − α2n+1||α2n|2 + |α2n−1α
2
2nα2n+1|
≤C(| − α2n−1 + α2n+1|+ |α2n|2 + |α2n−1|2 + |α2n+1|2).
By Proposition 2.3 α ∈ (K) =⇒ α ∈ `2
1. Therefore, an ∈ (G).
The proofs for bn are similar.
284 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
On a Class of Verblunsky Parameters
5. Inverse Geronimus Relations
Since there are four direct Szegő transforms (Geronimus relations), there are,
respectively, four inverse Szegő transforms (inverse Geronimus relations).
Proposition 5.1 (Inverse Geronimus Relations). Let the spectrum of J σ(J) ⊆
[−2, 2]. Let Pn and Qn be the monic orthogonal polynomials of the first and the
second kind, respectively, for J with the parameters an and bn. We define Fn(±2)
as follows:
• for the case (e)
Rn(−2) = Pn(−2), Rn(2) = Pn(2);
• for the case (o)
Rn(−2) = Pn(−2) +
Qn(−2)
m(−2)
, Rn(2) = Pn(2) +
Qn(2)
m(2)
;
• for the case (+)
Rn(−2) = Pn(−2), Rn(2) = Pn(2) +
Qn(2)
m(2)
;
• for the case (−)
Rn(−2) = Pn(−2) +
Qn(−2)
m(−2)
, Rn(2) = Pn(2).
We define
An = −Rn+1(−2)
Rn(−2)
, Bn =
Rn+1(2)
Rn(2)
.
Then αn for the inverse Geronimus relations are computed as follows:
• for the case (e)
α2n =
An −Bn
An + Bn
, α2n−1 = 1− An + Bn
2
;
• for the case (o)
−α2n+2 =
An −Bn
An + Bn
, −α2n+1 = 1− An + Bn
2
;
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 285
L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii
• for the case (+)
−α2n+1 =
An −Bn
An + Bn
, −α2n = 1− An + Bn
2
;
• for the case (−)
α2n+1 =
An −Bn
An + Bn
, α2n = 1− An + Bn
2
.
We define the “right” inverse Szegő transform by following Ryckman.
Definition 5.2. Let m(z) be the m-function of a Jacobi matrix J :
m(z) =
2∫
−2
ρ(x) dx
x− z
.
The “right” inverse Szegő transform for the J is
• (e) if both m(−2) and m(2) are infinite,
• (o) if both m(−2) and m(2) are finite,
• (+) if m(−2) is infinite and m(2) is finite,
• (−) if m(−2) is finite and m(2) is infinite.
R e m a r k 5.3. In what follows we will use the function
Fn(z) := Pn(z) +
Qn(z)
m(z)
.
In general,
Rn(±2) 6= Fn(±2). (5.1)
However, for the “right” inverse Szegő transform
Rn(±2) = Fn(±2).
Theorem 5.4. Let a, b be Jacobi parameters such that the corresponding
Jacobi matrix does not have a discrete spectrum, then
a, b ∈ (R) =⇒ α ∈ `2
1,
a, b ∈ (G) =⇒ α ∈ (K),
where α are defined by the “right” inverse Szegő transform.
286 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
On a Class of Verblunsky Parameters
R e m a r k 5.5. The first implication was proved by E. Ryckman in [7, 8]
(Cor. 5.8 below), the second is a result of this note. As it was shown in [5] for
the class (G) and in [7], for the class (R), a Jacobi matrix of the classes may have
at most finitely many eigenvalues outside [−2, 2]. To apply the inverse Szegő
transform we need to assume that there is no discrete spectrum.
R e m a r k 5.6. The following example from B. Simon’s book [12] (Example
13.1.3 Revisited on page 876) shows that the assertion of Theorem 5.4 may fail
if one chooses an inverse Szegő transform, which is not the “right” one. Namely,
let an = 1, bn = 0 for n ≥ 1. Then the spectral measure and m-function are
ρ(dx) =
√
4− x2
2π
χ[−2,2] dx, m(z) =
√
z2 − 4− z
2
.
Therefore, both m(±2) are finite. In this case the “right” inverse Szegő transform
is (o), and the corresponding αn = 0 is in (K).
For this example (e) is not the “right” inverse Szegő transform, the corre-
sponding sequence of Verblunsky parameters α2n = 0, α2n−1 = −(n + 1)−1,
n ≥ 0, is not in `2
1.
The key tool in proving Theorem 5.4 is the following asymptotics of the
so-called small solution of the Jacobi equation.
Theorem 5.7 (E. Ryckman [7, 8]). Let
Fn(z) = Pn(z) +
Qn(z)
m(z)
,
where Pn and Qn are as above. In other words, Fn(z) is a solution of the equation
Fk+1(z) + (bk+1 − z)Fk(z) + a2
kFk−1(z) = 0 (5.2)
with the initial conditions
F−1(z) = − 1
m(z)
, F0(z) = 1.
Let a, b ∈ (R), then
Fk(±2)
Fk−1(±2)
= ±1 + εk(±2), εk(±2) ∈ `2
1.
Corollary 5.8 (E. Ryckman [7, 8]). Let a, b ∈ (R), then
αn ∈ `2
1,
where αn are defined by the “right” inverse Szegő transform.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 287
L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii
P r o o f. Since we use the “right” inverse Szegő transform, (5.1) holds and
we have, by Theorem 5.7, that
An = −Rn+1(−2)
Rn(−2)
= −Fn+1(−2)
Fn(−2)
= 1 + `2
1
and
Bn =
Rn+1(2)
Rn(2)
=
Fn+1(2)
Fn(2)
= 1 + `2
1.
Hence, An −Bn and 1 − 1
2(An + Bn) are in `2
1. By the inverse Geronimus
relations we get that αn ∈ `2
1, (which is verified separately for even and odd
indices).
We will prove the second assertion in Theorem 5.4 as a corollary of the next
proposition.
Proposition 5.9. Let a, b ∈ (G), then
Fk(±2)
Fk−1(±2)
= ±1 + εk(±2), (5.3)
where ∞∑
k=0
k|εk(±2)− εk+1(±2)| < ∞.
P r o o f. We show it for z = 2; for z = −2 the proof is analogous. We use
again equation (5.2). From there we have
Fk+1(z)
Fk(z)
+ (bk+1 − z) + a2
k
Fk−1(z)
Fk(z)
= 0. (5.4)
We substitute (5.3) into (5.4) to get
(1 + εk+1) + (bk+1 − 2) + a2
k
1
1 + εk
= 0.
Transform it
(1 + εk+1) + (bk+1 − 2) + a2
k
(
1− εk +
ε2k
1 + εk
)
= 0,
(1 + εk+1) + (bk+1 − 2) +
(
1− εk +
ε2k
1 + εk
)
+ (a2
k − 1)
(
1− εk +
ε2k
1 + εk
)
= 0.
Since εk → 0, we get from here that
|εk+1 − εk| ≤ C(|bk+1|+ |εk|2 + |a2
k − 1|).
288 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
On a Class of Verblunsky Parameters
By assumption, a, b ∈ (G), then, by Proposition 3.3, a, b ∈ (R). Therefore, by
Theorem 5.7, εk ∈ `2
1 Hence, we get
∞∑
k=0
k|εk+1 − εk| < ∞.
P r o o f (of the second assertion in Theorem 5.4). As in Corollary 5.8
An = −Rn+1(−2)
Rn(−2)
= −Fn+1(−2)
Fn(−2)
= −1 + εn+1(−2)
and
Bn =
Rn+1(2)
Rn(2)
=
Fn+1(2)
Fn(2)
= 1 + εn+1(2).
Therefore,
An −An−1 = εn+1(−2)− εn(−2), Bn −Bn−1 = εn+1(2)− εn(2).
Hence, by Proposition 5.9,
∞∑
n=0
n(|An −An−1|+ |Bn −Bn−1|) < ∞.
Consider
∣∣∣∣
An+1 −Bn+1
An+1 + Bn+1
− An −Bn
An + Bn
∣∣∣∣ = 2
∣∣∣∣
An+1Bn −AnBn+1
(An+1 + Bn+1)(An + Bn)
∣∣∣∣
=2
∣∣∣∣
(An+1 −An)Bn −An(Bn+1 −Bn)
(An+1 + Bn+1)(An + Bn)
∣∣∣∣
≤C(|An+1 −An|+ |Bn+1 −Bn|).
Also
∣∣∣∣
(
1− An + Bn
2
)
−
(
1− An + Bn
2
)∣∣∣∣ ≤ C(|An+1 −An|+ |Bn+1 −Bn|).
Hence, by Inverse Geronimus relations we get that αn ∈ (K), (which is verified
separately for even and odd indices).
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 289
L. Golinskii, A. Kheifets, F. Peherstorfer, and P. Yuditskii
References
[1] L.D Faddeev, On the Relation Between S-Matrix and Potential for the One-
Dimensional Schrödinger Operator. — Dokl. Akad. Nauk SSSR 121 (1958). (Rus-
sian)
[2] P. Deift and E. Trubowitz, Inverse Scattering on the Line. — Comm. Pure Appl.
Math. 32 (1979), No. 2, 121–251.
[3] V.A. Marchenko, Operatory Shturma-Liuvillya i Ikh Prilozheniya. (Russian)
[Sturm–Liouville Operators and Their Applications] Izd-vo Naukova Dumka, Kiev,
1977.
[4] G.Sh. Guseinov, The Determination of an Infinite Jacobi Matrix from the Scattering
Data. — Soviet. Math. Dokl. 17 (1976), 596–600.
[5] G.Sh. Guseinov, The Scattering Problem for an Infinite Jacobi Matrix. — Izv.
Akad. Nauk ArmSSR. Ser. Mat. 12 (1977), No. 5, 365–379. (Russian)
[6] E. Ryckman, A Spectral Equivalence for Jacobi Matrices. — J. Approx. Theory
146 (2007), No. 2, 252–266.
[7] E. Ryckman, A Strong Szegő Theorem for Jacobi Matrices. — Comm. Math. Phys.
271 (2007), No. 3, 791–820.
[8] E. Ryckman, Erratum: A Strong Szegő Theorem for Jacobi Matrices. — Comm.
Math. Phys. 275 (2007), No. 2, 581–585.
[9] E. Ryckman, Two Spectral Equivalences for Jacobi Matrices. PhD Theses, UCLA,
Los Angeles, 2007.
[10] J. Geronimo and K. Case, Scattering Theory and Polynomials Orthogonal on the
Unit Circle. — J. Math. Phys. 20 (1979), No. 2, 299–310.
[11] J. Geronimo and K. Case, Scattering Theory and Polynomials Orthogonal on the
Real Line. — Trans. Amer. Math. Soc. 258 (1980), No. 2, 467–494.
[12] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory;
Part 2: Spectral Theory. AMS Colloquium Series, AMS, Providence, RI, 2005.
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