1. The estimates provide two-sided bounds on the sequences in terms of each other and the potential function of the operator."> 1. The estimates provide two-sided bounds on the sequences in terms of each other and the potential function of the operator.">
Estimates For Periodic Zakharov-Shabat Operators 2
Estimates For Periodic Zakharov-Shabat Operators 2
Estimates For Periodic Zakharov-Shabat Operators 2
Abstract
We consider the periodic Zakharov-Shabat operators on the real line. The spectrum of
this operator consists of intervals separated by gaps with the lengths |gn | > 0, n ∈ Z.
Let µ± n be the corresponding effective masses and let hn be heights of the corresponding
slits in the quasimomentum domain. We obtain a priori estimates of sequences g =
(|gn |)n∈Z , µ± = (µ± p
n )n∈Z , h = (hn )n∈Z in terms of weighted ℓ −norms at p > 1. The
proof is based on the analysis of the quasimomentum as the conformal mapping.
1
where I2 is the identity 2 × 2-matrix. Introduce the Lyapunov function ∆(z) = 12 Trψ(1, z).
Note that ∆(zn± ) = (−1)n , n ∈ Z, and the function ∆′ (z) has exactly one zero zn ∈ [zn− , zn+ ]
for each n ∈ Z. For each V there exists a unique conformal mapping (the quasi-momentum)
k : Z → K(h) such that (see [Mi1])
If the weight ωn = 1 for all n ∈ Z, then we will write ℓp0 = ℓp with the norm k · kp . For each
V we introduce the sequences
Z
1 2
h = (hn )n∈Z , γ = (|γn |)n∈Z, J = (Jn )n∈Z , Jn = |An | 2 > 0, An = v(z)dz > 0.
π γn
For the defocussing cubic non-linear Schrödinger equation (a completely integrable infinite
dimensional Hamiltonian system), An is an action variables (see [FM]). Recall the following
identities from [KK1]
Z ZZ
2 1 1
v(z)dz = |z ′ (k) − 1|2 dudv = kV k2 , (1.2)
π R π C 2
R1
where kV k2 = 0 (V12 (t) + V22 (t))dt. Korotyaev [K1] obtained the two-sided estimates for the
case ℓ2 , for example
kgk2 6 2khk2 6 πkgk2(2 + kgk22), (1.3)
1
√ kgk2 6 kV k 6 2kgk2(1 + kgk2). (1.4)
2
Our main goal is to obtain similar estimates in terms of the ℓp and ℓpω −norms. Our first
results are devoted to estimates in terms of ℓp − norms. Let below 1p + 1q = 1, p, q > 1.
Theorem 1.1. Let V ∈ L1 (0, 1). Then the following estimates hold true:
2(p+3)p
2−p kgkp 6 khkp 6 2kgkp(1 + αp0 kgkpp ), p ∈ [1, 2], αp0 = , (1.5)
π
2 2C p−1 2 2 π 2 1/p 1 1
p
khkp 6 Cp2 kgkq 1 + 2
kgk p−1
q , C p = , p > 2, + = 1, (1.6)
π π 2 p q
2
kgkp 2
6 kJkp 6 √ kgkp(1 + αp0 kgkpp )1/2 , p > 1, (1.7)
2 π
√
π
kJkp 6 khkp 6 4kJkp (1 + αp0 2p kJkpp ), p > 1. (1.8)
2
Estimates (2.1)-(2.4) are new for p ∈ [1, 2).
In the case zn− < zn+ we define the effective masses µ±
n by
(k − πn)2
z(k) − zn± = (1 + O(k − πn)) as z → zn± . (1.9)
2µ±
n
Theorem 1.2. Let h ∈ ℓpω , p ∈ [1, 2]. Then the following estimates hold true
n 1
o
± 0 p
khk∞ 6 min 2πkµ k∞ , kJkp,ω , 2kgkp,ω (1 + αp kgkp,ω ) , q (1.10)
1
kgkp,ω 6 2khkp,ω 6 c90 kgkp,ω , c0 = e π khk∞ , (1.11)
kgkp,ω 6 2kJkp,ω 6 c50 2kgkp,ω , (1.12)
√ r
π 5 π
kJkp,ω 6 khkp,ω 6 c0 kJkp,ω , (1.13)
2 2
kgkp,ω 6 2kµ± kp,ω 6 c18
0 kgkp,ω . (1.14)
Estimates (2.5)-(2.9) are new. Introduce the real Hilbert spaces ℓ2(m) , m ∈ R of the
P
sequences {fn }∞ 2
1 equipped with the norm kf k(m) = n>1 (2πn)
2m 2
fn . Korotyaev obtained
2
the two-sided estimates for the ℓ(m) − norms, m > 0 for the even case h−n = hn , n ∈ Z
([K2]-[K4]) and for ℓ21 −norms without symmetry ([K1], ([K6])) (in all these estimates the
factor c0 = ekhk∞ /π is absent).
khk∞ 6 kV k, (1.15)
2
kV k2 6 khkp kgkq , p > 1, (1.16)
π
2 2 2 2
kV k2 6 ( ) p khkpq kgkpp , p ∈ [1, 2], (1.17)
π
2 4
kV k2 6 khk∞ kgk1 6 2 kgk21 , (1.18)
π π
2
khk∞ 6 kgk1, kgk1 6 2khk1 . (1.19)
π
3
Note that the comb mappings are used in various fields of mathematics. We enumerate
the more important directions:
1) the conformal mapping theory, 2) the Löwner equation and the quadratic differentials,
3) the electrostatic problems on the plane, 4) analytic capacity, 5) the spectral theory of the
operators with periodic coefficients, 6) inverse problems for the Hill operator and the Dirac
operator, 7) the KDV equation and the NLS equation with periodic initial value problem.
Example of an electrostatic field. Consider the system of neutral conductors Γn , n ∈
Z on the plane for some h ∈ ℓpω . In other words, we embed the system of neutral conductors
Γn in the external homogeneous electrostatic field E0 = (0, −1) ∈ R2 on the plane. Then
on each conductor there exists the induced charge, positive en > 0 on the lower half of the
conductor Γn and negative (−en ) < 0 on the upper half of the conductor Γn , since their sum
equals zero. As a result we have new perturbed electrostatic field E ∈ R2 . It is well known
that E = iz ′ (k) = −∇y(k), k = u + iv ∈ K(h), z = x + iy, where z(k) is the conformal
mapping from K(h) onto the domain Z = C \ ∪gn . The function y(k) is called the potential
of the electrostatic field in K(h). The density of the charge on the conductor has the form
ρ(k) = |yu′ (k)|/4π, k ∈ Γn (see [LS]). Thus we obtain the induced charge en on the upper
half of the conductor Γ+ n = Γn ∩ C+ by:
Z
1 1
en = x′v (k)dv = |gn |.
4π Γ+n 4π
Introduce
R the bipolar moment dn of the conductor Γn with the charge density ρ(k) by
1
dn = 4π Γn
vxv (k)dv > 0. We transform this value into the form
Z
1 An
dn = v(x)dx = .
2π gn 4
In the paper [KK3] we study inverse problems for both the charge mapping h → e = (en )n∈Z
and the bipolar moment mapping h → J acting in ℓpω , p ∈ [1, 2]. In order to solve the inverse
problems we need a priori estimates from Theorems 1.1 and 1.2.
A priori estimates of potentials in terms of spectral data essentially simplify the proof in
the inverse problems. Such simplification was introduced by Garnett and Trubowitz [GT1]
and Kargaev and Korotyaev [KK2] and essentially was used in [K6]-[K9].
For the sake of the reader, we briefly recall the results existing in the literature about
2
the a priori estimates. Firstly, we describe a priori estimates for the Hill operators − dtd 2 + P
R 1
in L2 (R) with the real 1-periodic potential P ∈ L2 (0, 1) nd 0 P (t)dt = 0. The spec-
trum of this operator consists of intervals separated by gaps γn , n > 1 with the lengths
|γn | > 0, n ∈ Z. Marchenko and Ostrovski [MO1-2] obtained the estimates: kP k 6 C(1 +
R1
khk∞ )khk(1) , P khk(1) 6 CkP keCkP k for some absolute constant C, where kP k2 = 0 P 2 (t)dt
and khk2(1) = (2πn)2 h2n . These estimates are very rough since they used the Bernstein
inequality. Using the harmonic measure argument Garnett and Trubowitz [GT] obtained
kγk 6 (4 + khk(1) )khk(1) , where γ = (|γn |)∞
1 and recall that hn are heights on the quasimo-
mentum domain. First two-sided estimates (very rough) for g, h were obtained in [KK2].
4
Identities and a complete system of a priori estimates (in terms of gap lengths, effective
masses etc) were obtained by Korotyaev [K1]-[K6]. In the paper [K3], [K5], [K6] the following
estimates were obtained:
1 1
kγk2 6 6kP k(1 + kP k 3 ), kP k 6 4kγk2 (1 + kγk23 ),
1 1
2khk(1) 6 πkP k(1 + kP k 3 ), kP k 6 3(6 + khk∞ ) 2 khk(1) .
These estimates show the ”equivalence” of the values kγk, khk(1) , kP k. Note the author
solved the inverse problem and obtained the two-sided estimates for the case P = y ′, where
y ∈ L2 (0, 1). A priori two-sided estimates for the case P (m) ∈ L2 (0, 1), m > 1 were obtained
by Korotyaev in [K1].
Secondly for the Zakharov-Shabat systems the two-sided estimates were obtained by
Korotyaev for V ∈ L2 (0, 1) in [K1] (see (1.3)-(1.4)) and for V ′ ∈ L2 (0, 1) in [K6]. There are
no a priori estimates for the case V (m) ∈ L2 (0, 1), m > 2. The proof of a priori estimates in
[K1]-[K6] is based on the analysis of the quasimomentum as the conformal mapping.
We shortly describe the proof. In order to prove Theorem 1.1 -Proposition 1.3 we use the
analysis of a conformal mapping corresponding to quasimomentum of the Zakharov-Shabat
operator. That makes it possible to reformulate the problems for the differential operator as
the problems of the conformal mapping theory. Then we should study the metric properties
of a conformal mapping from C+ onto a ”comb” K+ (h). A similar analysis was done partially
in [KK1-3], [K1-3]. In the present paper we use an approach, based on the identities for the
Dirichlet integral (1.2) from [KK1] and the estimates from Theorem 2.7 and Lemma 3.1. We
emphasize the important role of the Dirichlet integral (this is the energy for the conformal
mapping) in this consideration.
We now describe the plan of the paper. In Section 2 we shall obtain some preliminaries
results and ”local basic estimates” in Theorem 2.7. In Section 3 we shall prove Lemma 3.1
and the main theorems. Moreover, we consider some examples, which describe our estimates.
2 Preliminaries
Consider a conformal mapping z : K+ (h) → C+ with asymptotics z(iv) = iv(1 + o(1)) as
v → ∞, where k = u + iv ∈ K(h). Here h = (hn )n∈Z ∈ ℓ∞ , hn > 0 is some sequence and
K+ (h) = C+ ∩ K(h) is the so-called comb domain, where K(h) is given by
K(h) = C \ ∪n∈Z Γn , Γn = [un − ihn , un + ihn ], u∗ = inf (un+1 − un ) > 0,
n
5
The function u(z) = Re k(z) is strongly increasing on each band σn and u(z) = un for all
z ∈ [zn− , zn+ ], n ∈ Z; the function v(z) = Im k(z) equals zero on each band σn and is strongly
convex on each gap gn 6= ∅ and has the maximum at some point zn given by v(zn ) = hn . If the
gap is empty we set zn = zn± . The function z(·) has an analytic extension (by the symmetry)
from the domain K+ (h) onto the domain K(h) and z(·) : K(h) → z(K(h)) = Z = C \ ∪g n
is a conformal mapping. These and others properties of the comb mappings it is possible to
find in the papers of Levin [Le].
We formulate our second result about the estimates for conformal mappings.
Theorem 2.1. Let u∗ = inf n (un+1 − un ) > 0. Then the following estimates hold true
(2 + π)p 2p(p+2)
khkp 6 2kgkp(1 + αp kgkpp), p ∈ [1, 2], αp = , (2.1)
πup∗
2 2 h 2C i p−1 2 2 π 2 1/p 1 1
p p−1
khkp 6 Cp kgkq 1 + kgkq , Cp = , p > 2, + = 1, (2.2)
π πu∗ 2 p q
kgkp 2
6 kJkp 6 √ kgkp (1 + αp kgkpp )1/2 , p > 1, (2.3)
2 π
√
π
kJkp 6 khkp 6 4kJkp (1 + αp 2p kJkpp ), p > 1. (2.4)
2
We formulate our second result about the estimates for conformal mappings.
Theorem 2.2. Let h ∈ ℓpω , p ∈ [1, 2] and let u∗ > 0. Then the following estimates hold true
−1
khk∞ 6 min{2πkµ± k∞ , kJkp,ω , 2π p kgkp,ω (1 + αp kgkpp,ω )1/q }, (2.5)
where k = u + iv, z = x + iy. The last identity holds since the Dirichlet integral is invariant
under the conformal mappings. In order to prove our main theorems we need the following
6
Proposition 2.3. Let h ∈ ℓ∞ and let u∗ > 0. Then
khk2∞
6 Q0 , (2.10)
2
πQ0 6 khkp kgkq , p > 1, (2.11)
2 2
ID 6 ( ) p khk2/q 2/p
p kgkp , p ∈ [1, 2], (2.12)
π
2
πQ0 6 khk∞ kgk1 6 kgk21, (2.13)
π
2
khk∞ 6 kgk1, kgk1 6 2khk1 . (2.14)
π
Proof of Theorem 1.1- Proposition 1.3 follow directly from Theorem 2.1- Proposition
2.3 and the identity (1.2).
We recall needed results. Below we will use very often the following simple estimate
y(k, e
h) > y(k, h), all k ∈ K+ (h), (2.18)
Q0 (e
h) 6 Q0 (h) and if Q0 (e
h) = Q0 (h), then e
h = h, (2.19)
|σn (e
h)| > |σn (h)|. (2.20)
Define the effective masses νn in the plane K(h) for the end of the slit [un + ihn , un −
ihn ], hn > 0 by
(z − zn )2
k(z) − (un + ihn ) = (1 + O(z − zn ))) as z → zn . (2.21)
2iνn
Thus we obtain νn = 1/|k ′′(zn )|, if hn > 0 and we set νn = 0 if |gn | = 0. We show the
possibility of the Lindelöf principle in the following Lemma.
7
Lemma 2.4. For each h ∈ ℓ∞ the estimate (2.10) and the following estimate hold true
νn 6 hn , all n ∈ Z. (2.22)
Then asymptotics (2.21) of the function z(k, h) as k → u0 + ih0 yields (2.22). In order to
prove (2.10) we use (2.19) since Q0 (e
h) = h20 /2. Note that it and (2.19) yield (2.10).
We recall estimates from [K4].
Theorem 2.5. Let h ∈ ℓ∞ . Then for any r > 0, n ∈ Z the following estimate holds true
π n h o ZZ
2 n
hn 6 max 1, |z ′ (k) − 1|2 dudv, Sr = {z ∈ C : | Re z| < r}, (2.23)
4 r un +Sr
π π2 n khk o π2 n I 1/2 o
∞
ID 6 khk22 6 max 1, ID 6 max 1, D ID , (2.24)
4 2 u∗ 2 u∗
1 2
kgk2 6 khk2 6 πkgk2 1 + 2 kgk22 , (2.25)
2 u∗
√
kgk2 √ 2
6 kJk2 6 2kgk2 (1 + kgk). (2.26)
2 u∗
In order to prove Theorem 2.7 we need the following result about the simple mapping
and the domain Sr = {z ∈ C : | Re z| < r}, r > 0.
√
Lemma 2.6. The function f (k) = k 2 + h2 , k ∈ C \ [−ih, ih], h > 0 is the conformal
mapping from C \ [−ih, ih] onto C \ [−h, h] and Sr \ [−h, h] ⊂ f (Sr \ [−ih, ih]) for any r > 0.
Proof. Consider the image of the half-line k = r + iv, v > 0. We have the equations
x2 + y 2 = ξ ≡ r 2 + h2 − v 2 , xy = rv. (2.27)
The second identity in (2.27) yields x > 0 since y >p0 . Then x4 − ξx2 − r 2 v 2 = 0, and
enough to check the following inequality x2 = 21 (ξ + ξ 2 + 4r 2v 2 ) > r 2 . The last estimate
follows from the simple relations
(r 2 + h2 − v 2 )2 + 4r 2v 2 > (r 2 + v 2 − h2 )2 , 4r 2 v 2 > 4r 2 (v 2 − h2 ).
We prove the local estimates for the small slits, which are crucial for us.
8
Theorem 2.7. Let h ∈ ℓ∞ . Assume that (un − r, un + r) ⊂ (un−1, un+1 ) and hn 6 2r , for
some n ∈ Z and r > 0. Then
ZZ
± 2+π ± p 1
|hn − |µn || 6 |µn | In , In = |z ′ (k) − 1|2 dudv, (2.28)
r π un +Sr
2+π p
0 6 hn − νn 6 2 hn In , (2.29)
r
|gn | 2+π p
0 6 hn − 6 hn In . (2.30)
2 r
p
Proof. Define the functions f (k) = k 2 + h2n , k ∈ Sr \ [−ihn , ihn ], φ = f −1 and F (w) =
z(un + φ(w), h), w = p + iq, where the variable w ∈ G1 = f ((Sr \ [−ihn , ihn ])). The function
F is real for real w, then F is analytic in the domain G = G1 ∪ [−hn , hn ] and Lemma 2.6
yields Sr ⊂ G. Let now |w1 | = 2r and Br = {z : |z| < r}. Then the following estimates hold
ZZ
√ r ′
π |F (w1 ) − 1| 6 ( |F ′ (w) − 1|2 dpdq)1/2 6 (2.31)
2 Br
ZZ 1/2 Z Z 1/2
′ 2 ′ 2
6 |(F (w) − φ(w)) | dpdq + |φ (w) − 1| dpdq .
Br Br
The invariance of the Dirichlet integral with respect to the conformal mapping gives
ZZ ZZ ZZ
′ 2 ′ 2
|(F (w)−φ(w)) | dpdq = |z (k)−1| dudv 6 |z ′ (k)−1|2 dudv = πIn . (2.32)
Br φ(Br ) Sr +un
2 π p 2 + πp
|F ′ (w1 ) − 1| 6 (1 + ) In = In , (2.34)
r 2 r
and the maximum principle yields the needed estimates for |w1 | 6 r/2.
We prove (2.28) for µ+ ±
n . The definition of µn (see (1.9)) implies
g(x) x hn
F ′ (hn ) = lim z ′ (un + g(x)) · g ′(x) = lim +
· = +.
xցhn xցhn µn g(x) µn
9
The substitution of the last identity into (2.34) gives (2.28). The proof for µ−
n is similar.
We show (2.29). The definition of νn (see (2.21)) yields
Hence
π2 u∗ πp u∗
hn 6 In , if hn > , and hn 6 In , if hn 6 . (3.2)
u∗ 4 2 4
Moreover, (2.30) yields
|gn | 2+π p u∗
hn 6 +2 hn In , if hn < , (3.3)
2 u∗ 4
and then
2+π u∗
hn 6 2π In , if hn < , |gn | 6 hn , (3.4)
u∗ 4
10
π
√
since hn 6 2
In . Hence using (3.4), (3.2), we obtain
2+π 2+π
if |gn | 6 hn ⇒ hn 6 2π In = C1 In , C1 = 2π .
u∗ u∗
The last inequality and (2.10) yield
X 1 X 1 1 p+1
1 p+1
Conversely, if we assume that kgkp 6 C1p ID2p , then (3.5), (2.12) implies
1 h 2 p2 2 i p+1
2p
khkp 6 2C1 p
khkp kgk2/p
q
p .
π
Hence
h 1 i p+1
2 p2 2
khk1/p
p
p
6 2C1 (2/π)kgkp ⇒ khkp 6 2p C1p (2/π)p+1 kgk1+p
p ,
Consider the case b > 1. Then the substitution of (2.10), (2.11) into (3.6) yield
p+1 p+1
khkp 6 Cp ID2p u−1/p
∗ 6 u−1/p
∗ Cp [(2/π)khkp kgkq ] 2p ,
and
p−1 p+1 2p p+1 p+1
khkp2p 6 Cp u−1/p
∗ [(2/π)kgkq ] 2p ⇒ khkp 6 (Cp u∗−1/p ) p−1 (2/π) p−1 kgkqp−1 ,
11
recall that αp = (2p+2(2 + π)/u∗)p /π. Inequality Jn2 6 4h2n /π (see (2.17)) yields the first
estimate in (2.4). Using (2.1) and kgkp 6 2kJkp (see (2.3)) we deduce that
for all x ∈ gn = (zn− , zn+ ). In order to prove Theorem 2.2 we need the following results.
khk∞
Lemma 3.1. Let h ∈ ℓ∞ and u∗ > 0 and c = e . Then the following estimates hold:
u∗
πs n 5π
o
2
s = inf |σn | 6 u∗ 6 max e , c 2 , (3.8)
2
2khk∞
1+ 6 c9 , (3.9)
sπ
2khk∞
max Yn (x) 6 , n ∈ Z, (3.10)
n∈gn πs
2khk∞
2hn 6 |gn |(1 + max Yn (x)) 6 |gn |(1 + ) 6 |gn |c9 , n ∈ Z. (3.11)
n∈gn πs
Proof. Introduce the domain
u∗ u∗
G = {z ∈ C : h+ > Im z > 0, Re z ∈ (− , )} ∪ {Im z > h+ }, h+ = khk∞ .
2 2
Let F be the conformal mapping from G onto C+ , such that F (iy) ∼ iy as y ր +∞ and
let α, β be images of the points u2∗ , u2∗ + ih+ respectively. Define the function f = Im F . Fix
any n ∈ Z. Then the maximum principle yields
1
y(k) = Im z(k, h) > f (k − pn ), k ∈ G + pn , pn = (un−1 + un ).
2
Due to the fact that these positive functions equal zero on the interval (pn − u2∗ , pn + u2∗ ), we
obtain
u∗ u∗
∂v y(x) = ∂u x(x) > ∂v f (x − pn ), x ∈ (pn − , pn + ).
2 2
∂
where ∂x = ∂x . Then
u
Z∗ /2
z(un ) − z(un−1 ) > ∂v f (x) dx = 2α > 0,
−u∗ /2
12
and the estimate s 6 u∗ (see [KK1]) implies 2α 6 s 6 u∗ . Let w : C+ → G be the inverse
function for F , which is defined uniquely and the Christoffel-Schwartz formula yields
Z zr 2
t − β2
w(z) = dt, 0 < α < β.
0 t2 − α2
Then we have Z r Z r
α β
u∗ β 2 − t2 β 2 − t2
= dt, h+ = dt. (3.12)
2 0 α2 − t2 α t2 − α2
The first integral in (3.12) has the simple double-sided estimates
Z α Z α
u∗ βdt βπ
α= dt 6 6 √ = ,
0 2 0
2
α −t 2 2
that is
2α 6 s 6 u∗ 6 πβ. (3.13)
Consider the second integral in (3.12). Let ε = β/α > 5 and using the new variable
t = α cosh r, cosh δ = ε, we obtain
s
Z δp Z δ/2
cosh2 r 2
h+ = α ε2 − cosh2 rdr > αε 1− 2
dr > βδ ,
0 0 ε 5
cosh2 r
2 6 e−δ (1 + e−δ )2 6 ε−1(1 + ε−1 )2 .
cosh δ
Due to ε 6 eδ we get ε 6 exp(5h+ /2β) and estimate (3.13) implies
1 π 5π
6 exp( h+ ), if ε > 5. (3.14)
s 2u∗ 2u∗
If ε 6 5, then using (3.13) again we obtain
1 ε πε π
6 6 6 5, if ε 6 5.
s 2β 2u∗ 2u∗
Using (3.7), (3.9) and simple inequality vn (zn ) 6 |gn |/2 we have (3.11).
We prove the two-sided estimates of hn , |gn |, µ±
n , Jn in the weight spaces.
13
±
Proof of Theorem 2.2. The first estimate in √ (2.5) follows from hn 6 2π|µn | (see [KK1]).
The second one in (2.5) follows from khk∞ 6 ID = kJk √2 6 kJkp 6 kJkp,ω since ωn > 1 for
any n ∈ Z. Moreover, substituting (2.12) into khk∞ 6 ID , using (2.1) and kf kp 6 kf kp,ω
for any f , we obtain the last estimate in (2.5).
Recall that c = exp khk u∗
∞
. The first estimate in (2.6) follows from (2.15). Due to (3.11)
9
we get 2hn 6 c |gn |, which yields the second estimate in (2.6).
The first estimate in (2.7) follows from (2.17). Using (2.17), (3.11) we have Jn2 6
2|gn |hn /π 6 (c9 /π)|gn |2 , which gives the second estimate in (2.7).
The first estimate in (2.8) follows from (2.17), (2.15). Using (2.17), (3.11) we obtain
h2n 6 c9 |gn |hn /2 6 (πc9 /2)Jn2 , which yields the second inequality in (2.8).
Identity 2|µ± ± 2 ±
n | = |gn |[1 + Yn (zn )] (see [KK1]) implies 2|µn | > |gn |, which yields the first
inequality in (2.9). Moreover, using (3.11) we obtain the estimate 2|µ± 18
n | 6 c |gn |, which
gives the second one in (2.9).
Recall that for a compact subset Ω ⊂ C the analytic capacity is given by
h i
C = C(Ω) = sup |f ′ (∞)| : f is analytic in C \ Ω; |f (k)| 6 1, k ∈ C \ Ω , (3.15)
where f ′ (∞) = lim k(f (k) − f (∞)). We will use the well known Theorem (see [Iv], [Po])
|k|→∞
Theorem ( Ivanov-Pommerenke ). Let E ⊂ R be compact. Then the analytic capacity
C(E) = |E|/4, where |E| is the Lebesgue measure (the length) of the set E. Moreover, the
Ahlfors function fE (the unique function, which gives sup in the definition of the analytic
capacity) has the following form:
Z
exp ( 12 φE (z)) − 1 dt
fE (z) = 1 , φE (z) = ; z ∈ C \ E. (3.16)
exp ( 2 φE (z)) + 1 E z −t
We will use the following simple remark: Let S1 , S2 , . . . , SN be disjoint continua in the plane
C; D = C\∪N ′
n=1 Sn . Introduce the class Σ (D) of the conformal mapping w from the domain
D onto C with the following asymptotics: w(k) = k + [Q(w) + o(1)]/k, k → ∞. If Ω ⊂ C is
compact; D = C \ Ω, g ∈ Σ′ (D), then C(Ω) = C(C \ g(D)). It follows immediately from the
definition of the analytic capacity.
Let ℓ2f in ⊂ ℓ2 be the subset of finite sequences of non negative numbers. Then, using the
Ivanov-Pommerenke Theorem and the last remark we obtain
kg(h)k1 = C(Γ(h)), where h ∈ ℓ2f in , Γ(h) = ∪[un − ihn , un + ihn ];
Now we estimate the Dirichlet integral ID (h) = 2Q0 (h) for the case u∗ > 0, using a
geometric construction.
Theorem 3.2. Let h ∈ ℓ∞ , hn → 0 as |n| → ∞; and let e h=eh(h) be given by
e
if h = 0, then h = 0,
if h 6= 0, let an integer n1 be such that e hn1 = hn1 = maxn∈Z hn > 0; assume that the
numbers hn1 , hn2 , . . . , hnk are defined, then nk+1 is given by
e
hnk+1 = hnk+1 = max hn > 0, B = {n ∈ Z : |un − uns | > hns , 1 6 s 6 k}, (3.17)
n∈B
14
and let e
hn = 0, if n ∈
/ {nk , k ∈ Z}.
Then the following estimates hold:
√
1 e 2 ID (h) 2 2 e 2
khk2 6 Q0 (h) = 6 khk2 . (3.18)
π2 2 π
Proof. The Lindelöf principal yields Q0 (e h) 6 Q0 (h). On the other hand open squares
Pk = (unk − tk , unk + tk ) × (−tk , tk ), tk ≡ hnk = e hnk , k ∈ Z, does not overlap. Then applying
e
(2.23) to the function (z(k, h) − k) and Pk , we obtain
Z Z
2
2tk 6 π |z ′ (k, e
h) − 1|2 dudv, (3.19)
Pk
and
1 1 X 2 1
Q0 (e
h) = ID (e
h) > 2 tk = 2 ke
hk22
2 π k>1 π
References
[FM] Flaschka H., McLaughlin D. Canonically conjugate variables for the Korteveg- de Vries equation
and the Toda lattice with periodic boundary conditions. Prog. of Theor. Phys. 55(1976), 438-456.
[GT] Garnett J., Trubowitz E.: Gaps and bands of one dimensional periodic Schrödinger operators.
Comment. Math. Helv. 59(1984), 258-312.
[GT1] Garnett J., Trubowitz E.: Gaps and bands of one dimensional periodic Schrödinger operator II.
Comment. Math. Helv. 62(1987), 18-37.
[Iv] Ivanov L.D. On a hypothesis of Denjoy, Usp. Mat. Nauk, 18(1963), 4(112), 147-149.
[J] Jenkins A.: Univalent functions and conformal mapping. Berlin, Göttingen, Heidelberg: Springer,
1958.
15
[JM] R. Johnson; J. Moser. The rotation number for almost periodic potentials. Commun. Math. Phys.
84(1982), 403-430.
[KK1] Kargaev P.; Korotyaev E. Effective masses and conformal mappings. Commun. Math. Phys.
169(1995), 597-625
[KK2] Kargaev P., Korotyaev E. The inverse problem for the Hill operator, a direct method. Invent. Math.
129(1997), 567-593.
[KK3] Kargaev P., Korotyaev E. Inverse electrostatic problems on plane, in preparation.
[K1] Korotyaev E. Metric properties of conformal mappings on the complex plane with parallel slits.
Inter. Math. Reseach. Notices. 10(1996), 493-503.
[K2] Korotyaev E. : Estimates for the Hill operator.I, Journal Diff. Eq. 162(2000), 1-26.
[K3] Korotyaev E. Estimate for the Hill operator.II, J. Differential Equations 223 (2006), 229–260.
[K4] Korotyaev E. The estimates of periodic potentials in terms of effective masses. Commun. Math.
Phys. 183(1997), 383-400.
[K5] Korotyaev E. Estimate of periodic potentials in terms of gap lengths. Commun. Math. Phys.
197(1998), 521-526.
[K6] Korotyaev E. Inverse oroblem and estimates for periodic Zakharov-Shabat systems, J. Reiner Angew.
Math. 583(2005), 87-115.
[K7] Korotyaev E. Characterization of the spectrum of Schrödinger operators with periodic distributions.
Int. Math. Res. Not. 37(2003), 2019–2031.
[K8] Korotyaev, E. Inverse problem for periodic ”weighted” operators. J. Funct. Anal. 170 (2000), 188–
218.
[K9] Korotyaev, E. The inverse problem for the Hill operator. I. Internat. Math. Res. Notices 1997, no.
3, 113–125.
[LS] Lavrent’ev M., Shabat B.: Methoden der komplexen Funktionentheorie, (U. Pirl, R. Kühnua, and
Wolfersdorf, eds.) VEB, Deutscher Verlag der Wissenschaften, Berlin, 1967.
[Le] Levin B.: Majorants in the class of subharmonic functions.1-3. Theory of functions, functional
analysis and their applications. 51(1989), 3-17 ; 52(1989), 3-33 . Russian.
[MO1] Marchenko V., Ostrovski I.: A characterization of the spectrum of the Hill operator. Math. USSR
Sb. 26(1975), 493-554 .
[MO2] Marchenko V., Ostrovski I.: Approximation of periodic by finite-zone potentials. Selecta Math.
Sovietica. 6(1987), No 2, 101-136.
[Po] Pommerenke C. Boundary behaviour of conformal maps, Berlin, Springer-Verlag, 1992.
16