Stability for the inverse resonance problem for the CMV operator
aa r X i v : . [ m a t h . SP ] J a n Proceedings of Symposia in Pure Mathematics
Stability for the inverse resonance problem for the CMVoperator
Roman Shterenberg, Rudi Weikard, and Maxim Zinchenko
Dedicated with great pleasure to Fritz Gesztesy on the occasion of his 60th birthday.
Abstract.
For the class of unitary CMV operators with super-exponentiallydecaying Verblunsky coefficients we give a new proof of the inverse resonanceproblem of reconstructing the operator from its resonances - the zeros of theJost function. We establish a stability result for the inverse resonance problemthat shows continuous dependence of the operator coefficients on the locationof the resonances.
1. Introduction
In recent years there has been a substantial interest in inverse resonance prob-lems for operators that arise in mathematical physics. An inverse resonance prob-lems asks whether an operator is determined by its resonances. One is typicallyinterested in reconstructing coefficients of an operator from a given set of reso-nances and in stability of the solution of the inverse resonance problem under smallperturbation of the resonances. The case of Schr¨odinger and Jacobi operatorshave received most of the attention, see for instance [ , , – ] and the referencestherein.The inverse resonance problem for a class of unitary operators known as CMVoperators has been considered in [ ]. These operators are represented by five-diagonal infinite unitary matrices giving rise to unitary operators on ℓ ( N ). Theyare closely connected with trigonometric moment problems and with orthogonalpolynomials and finite measures on the unit circle. The entries in a CMV matrixare determined by the Verblunsky coefficients, a sequence of complex numbers inthe unit disk. For background information on CMV matrices we refer the readerto Simon’s two-volume monograph [ , ].In this paper we continue our study of the inverse resonance problem for unitaryCMV operators associated with sequences of exponentially decaying Verblunsky Mathematics Subject Classification.
Key words and phrases.
Inverse problem, eigenvalues and resonances, Jost function, CMV.The authors were supported in part by NSF grants DMS-0901015, DMS-0800906, and DMS-0965411, respectively. c (cid:13) coefficients. For these operators we review the concept of a resonance and give asimplified proof of the uniqueness result in [ ] that the location of all resonancesdetermines the Verblunsky coefficients uniquely. The main result of the presentstudy is a stability result: Suppose two CMV operators in our class have resonances z n and ˘ z n , respectively. Further suppose that these resonances are respectively closeto each other as long as they are not too large. Then the Verblunsky coefficients arerespectively close to each other, too. We will now make these statements precise.Throughout this paper γ > η >
0, and
Q > B ( γ, η, Q ) as the set of those sequences of Verblunsky coefficients α : N → D satisfying the following two conditions:(1) | α k | ≤ η exp( − k γ ) for all k ∈ N and(2) Q ∞ j =1 (1 − | α j | ) ≥ /Q .We note that the above two conditions are essentially independent as the first oneenforces the decay of | α k | for large k and the second one bounds | α k | away from 1for small k . Also, two sets { z , ..., z N } and { ˘ z , ..., ˘ z N } of complex numbers will becalled respectively ε -close if | z n − ˘ z n | < ε for all n ∈ { , ..., N } .The following technical result required for our proof of stability for the inverseresonance problem is of independent interest. We will prove it in Section 4.2. Theorem . Suppose α is a sequence of Verblunsky coefficients in B ( γ, η, Q ) and U is the associated CMV operator. Then there is a positive number δ such that U has no resonances in the disk { z : | z | < δ } . The main purpose of this paper is to prove the following theorem, which wewill do in Section 4.
Theorem . Suppose α and ˘ α are two sequences of Verblunsky coefficientsin B ( γ, η, Q ) and U and ˘ U are the associated CMV operators. Let δ be the numberintroduced in the previous theorem. Further suppose that, for two numbers R > and ε ∈ (0 , δ/ , the resonances of U and ˘ U in the circle | z | < R , if there are any,are respectively ε -close. Then there is a constant A , depending only on γ , η , and Q , such that | α n − ˘ α n | ≤ A (6 Q ) n (cid:18) ε + (log R ) γ/ ( γ − R (cid:19) for all n ∈ N . Theorem 1.2 extends earlier results on stability of the inverse resonance prob-lems for Schr¨odinger and Jacobi operators [ – ] to the case of unitary CMVoperators. In this note we present a new approach to the stability of the inverseresonance problems. Unlike the earlier work our approach does not rely on a heavymachinery of the transformation operators but instead uses the Schur algorithm - asimple recursion relation that arises naturally in the context of CMV operators. Wepoint out that this approach is not specific to CMV operators only. There is alsoa similar simple recursive approach to the stability result of the inverse resonanceproblem for Jacobi operators.The paper is organized as follows. In Section 2 we introduce the basics of CMVoperators, define the main objects, and state some known facts that are central toour study. In Section 3 we set the stage for the stability result and give a new proofof the inverse resonance problem for CMV operators that first appeared in [ ]. TABILITY FOR THE INVERSE RESONANCE PROBLEM FOR THE CMV OPERATOR 3
Section 4 is devoted to the stability of the inverse resonance problem and containsthe proof of our main Theorem 1.2.Notation: In the following, we denote the set of all complex-valued sequencesdefined on N by C N . The Hilbert space of all square summable complex-valuedsequences is ℓ ( N ) and its scalar product h· , ·i is linear in the second argument.Recall that the vectors δ k ∈ ℓ ( N ), k ∈ N , defined by the requirement that δ k ( n )equals Kronecker’s δ k,n , form the standard basis in ℓ ( N ). The open unit disk inthe complex plane is denoted by D .
2. Preliminaries2.1. The CMV equations and the CMV operator.
The CMV equationsare defined through a sequence of coefficients α : N → D ; these are called Verblunskycoefficients. For z ∈ C \{ } the CMV equations are the recursive equations (cid:18) uv (cid:19) ( z, k ) = T ( z, k ) (cid:18) uv (cid:19) ( z, k − , k ∈ N (2.1)where, using the abbreviation ρ k = p − | α k | , T ( z, k ) = ρ k α k z /z α k ! , k odd , ρ k α k α k ! , k even . It is clear that the space of solutions of these equations is 2-dimensional. A basis ofsolutions is given by the sequences ϑ ( z, · ) and ϕ ( z, · ) defined by the initial conditions ϑ ( z,
0) = ( − , ⊤ and ϕ ( z,
0) = (1 , ⊤ . (2.2)Clearly ϑ ( · , k ) and ϕ ( · , k ) are analytic in C \{ } for any k ∈ N . Since T (1 /z, k ) = (cid:18) (cid:19) T ( z, k ) (cid:18) (cid:19) ( v, u ) ⊤ (1 /z, · ) satisfies the CMV equations if ( u, v ) ⊤ ( z, · ) does. This implies, takingthe initial conditions into account, that ϕ ( z, k ) = (cid:18) (cid:19) ϕ (1 /z, k ) and ϑ ( z, k ) = − (cid:18) (cid:19) ϑ (1 /z, k ) (2.3)whenever z ∈ C \{ } .To define the CMV operator set firstΘ k = (cid:18) − α k ρ k ρ k α k (cid:19) . These blocks are then used to define W = L ∞ k =1 Θ k − and V = 1 ⊕ ( L ∞ k =1 Θ k )where 1 is interpreted as a 1 × V W is denoted by U and is called the CMV operator. U , V , and W are defined on C N (and map to thatspace). Their restrictions to ℓ ( N ) are unitary operators which we denote usingthe same letters as the precise meaning will always be clear from the context.The following lemma was established in [ ]: ROMAN SHTERENBERG, RUDI WEIKARD, AND MAXIM ZINCHENKO
Lemma . Suppose z ∈ C \{ } . ( u, v ) ⊤ ( z, · ) is a solution of the CMV equa-tions (2.1) if and only if (cid:18) U V (cid:19) (cid:18) uv (cid:19) ( z, · ) = [ u ( z, · ) + ( v ( z, − u ( z, δ ] (cid:18) z (cid:19) . Hence if u ( z, · ) is the first component of ϕ ( z, · ) we have U u = zu .Now suppose that | z | 6 = 1 so that z is in the resolvent set of the unitary operator U and set u ( z, · ) = 2 z ( U − z ) − δ which is in ℓ ( N ). Define m ( z ) = 1 + u ( z,
0) = h δ , ( U + z )( U − z ) − δ i (2.4)which is analytic in D , and, assuming also z = 0, ω ( z, · ) = ϑ ( z, · ) + m ( z ) ϕ ( z, · ) . Both components of ω ( z, · ) are square summable (the first component is u and thesecond is V − ( u + 2 δ )). Moreover, ω ( z, · ) and its constant multiples are the onlysquare summable solutions of the CMV equations (2.1) since the unitary operator U can not have eigenvalues away from the unit circle. Thus, employing (2.3), wefind that − (cid:18) (cid:19) ω (1 /z, · ) = ϑ ( z, · ) − m (1 /z ) ϕ ( z, · )is equal to ω ( z, · ) which implies that m ( z ) = − m (1 /z ).The function m is called the Weyl-Titchmarsh m -function while the sequence ω ( z, · ) is called the Weyl-Titchmarsh solution of the CMV equations (2.1). We alsonote that, for z = 0 we get u (0 , · ) = 0, m (0) = 1, and v (0 , · ) = 2 δ so that, forevery k ∈ N , the singularity of ω ( · , k ) at 0 is removable.It follows from (2.4) via the spectral theorem that m ( z ) = I ∂ D ζ + zζ − z dµ ( ζ ) = 12 π Z π − π e it + z e it − z d ˜ µ ( t ) , where dµ denotes the spectral measure associated with the unitary operator U andthe cyclic vector δ and ˜ µ ( t ) = 2 πµ ( e it ) gives rise to the corresponding measureon [ − π, π ]. The case z = 0 shows that dµ is a probability measure. Since (e it + z ) / (e it − z ) has positive real part for all z ∈ D , it follows that m is a Caratheodoryfunction, that is, m is analytic on D , m (0) = 1, and Re m ( z ) > | z | < U − z ) − in (2.4) shows m ( z ) = 1 + 2 ∞ X n =1 z n h δ , U − n δ i . This implies m ( n ) (0) = 2 n ! h δ , U − n δ i so that m ′ (0) = − α and m ′′ (0) = 4 ¯ α − ρ ¯ α . (2.5) Assuming super-exponential decay of the Verblunsky coefficients, i.e., | α k | ≤ η exp( − k γ )Jost solutions of the CMV equations were defined and investigated in [ ]. Werepeat here briefly the most important results. TABILITY FOR THE INVERSE RESONANCE PROBLEM FOR THE CMV OPERATOR 5
Defining ζ k = ( z, k odd1 , k evenit was proved in [ ] that the Volterra-type equations F ( z, k ) = (cid:18) (cid:19) − ∞ X n = k +1 (cid:18) α n ζ n z n − k − α n ζ k +1 (cid:19) F ( z, n ) , k ∈ N , (2.6)have a unique solution for any complex number z . Either component of F ( · , k ) isan entire function of growth order zero and, if | z | ≥ k F ( z, k ≤ exp( η + 2 K ( z ) log | z | ) ∞ Y n =1 (1 + | α n | ) (2.7)where k · k denotes the 2-norm in C and K ( z ) = ⌊ (log 2 | z | ) γ − ⌋ . We also recall that k F ( z, k ) − (cid:18) (cid:19) k ≤ β ( z, k ) exp( β ( z, k )) (2.8)where β ( z, k ) = P ∞ n = k +1 | α n | max { , | z | n − } .Setting C k = Q ∞ j = k +1 ρ − j , it is straightforward to show that the sequence ν ( z, · )defined by ν ( z, k ) = 2 z ⌈ k/ ⌉ C k (cid:18) (cid:19) k +1 F ( z, k ) (2.9)satisfies the CMV equations (2.1) as does the sequence˜ ν ( z, k ) = (cid:18) (cid:19) ν (1 /z, k ) . If | z | < ν ( z, · ) are in ℓ ( N ) so that ν ( z, · ) must be a multipleof the Weyl-Titchmarsh solution ω ( z, · ), i.e., ν ( z, · ) = ψ ( z ) ω ( z, · ) = ψ ( z )( ϑ ( z, · ) + m ( z ) ϕ ( z, · )) , | z | < ψ . Evaluating at k = 0 using (2.2) and (2.9) yields ψ ( z ) = ( − , ν ( z, C (1 , − F ( z, , (2.10) ψ ( z ) m ( z ) = (1 , ν ( z, C (1 , F ( z, , (2.11)hence ψ and ψ m extend to entire functions. Consequently m extends to a mero-morphic function on C which we will denote by M (we emphasize that M ( z ) = m ( z )for | z | > ν we obtain the relationships ν ( z, · ) = ψ ( z )( ϑ ( z, · ) + M ( z ) ϕ ( z, · ))and ˜ ν ( z, · ) = ψ (1 /z )( − ϑ ( z, · ) + M (1 /z ) ϕ ( z, · ))which are valid for any z ∈ C \{ } . The solutions ν ( z, · ) and ˜ ν ( z, · ) are calledrespectively the Jost solutions of the CMV equations if | z | < | z | >
1. The
ROMAN SHTERENBERG, RUDI WEIKARD, AND MAXIM ZINCHENKO function ψ is called the Jost function. It is an entire function of growth order zero.Its zeros are called resonances.We end this section with the following observation. Since det T ( z, k ) = − ν ( z, k ) , ˜ ν ( z, k )) = ( − k det( ν ( z, , ˜ ν ( z, . The asymptotic behavior of ν ( z, k ) and ˜ ν ( z, k ) as k tends to infinity shows thatthe left hand side is equal to 4( − k +1 . If we now pick z on the unit circle so that z = 1 /z we obtain from this 1 = | ψ ( z ) | Re( M ( z )) . (2.12) Closely related to the concept of a Caratheodoryfunction is that of a Schur function, that is, a function defined and analytic on D whose modulus never exceeds one. Indeed if f is a Caratheodory function then( f − / ( f + 1) is a Schur function, while (1 + g ) / (1 − g ) is a Caratheodory functionif g is a Schur function. Note also that, by Schwarz’s lemma, z g ( z ) /z is a Schurfunctions if g is a Schur function and g ( z ) = 0.For k ∈ N and z ∈ D we define now the functionsΦ k ( z ) = 1 z (1 , ω ( z, k )(0 , ω ( z, k ) and Φ k +1 ( z ) = (0 , ω ( z, k + 1)(1 , ω ( z, k + 1) (2.13)and we note that in place of ω we may as well put ν since these are multiples ofeach other.The initial conditions satisfied by the Weyl-Titchmarsh solution ω show thatΦ ( z ) = 1 z m ( z ) − m ( z ) + 1 (2.14)which is a Schur function.Using the CMV equations (2.1) one may check thatΦ k ( z ) = 1 z S ( α k , Φ k − ( z ))where S ( w, · ) is the M¨obius transform z S ( w, z ) = z + w wz which maps D to D provided w ∈ D .Next, taking (2.5) into account, one sees that Φ (0) = m ′ (0) = − ¯ α . Thisshows that Φ is a Schur function and we find, again by (2.5), that Φ (0) =Φ ′ (0) / (1 − | α | ) = − ¯ α .Consider now the truncated sequence of Verblunsky coefficients n α N + n .The Jost solution for this problem is given by k z − N ν ( z, N + k ). Consequently,the function Φ N plays the same role for the truncated sequence as Φ plays forthe full sequence. Therefore we have Φ N (0) = − ¯ α N +1 and Φ N +1 (0) = − ¯ α N +2 .Thus any of the functions Φ k is a Schur function andΦ k (0) = − ¯ α k +1 . (2.15)The hyperbolic distance on D , given by d [ w , w ] = 2 tanh − (cid:12)(cid:12)(cid:12)(cid:12) w − w − w w (cid:12)(cid:12)(cid:12)(cid:12) = log 1 + | w − w | / | − w w | − | w − w | / | − w w | , TABILITY FOR THE INVERSE RESONANCE PROBLEM FOR THE CMV OPERATOR 7 is invariant under M¨obius transforms which map D onto itself. Hence, employing thetriangle inequality and the fact that, for | z | <
1, we have d [ z Φ k ( z ) , ≤ d [Φ k ( z ) , d [Φ k − ( z ) ,
0] = d [ z Φ k ( z ) , α k ] ≤ d [Φ k ( z ) ,
0] + d [0 , α k ] . Inequality (2.8) combined with (2.9) and (2.13) implies that Φ k ( z ) tends uniformlyto zero as k tends to infinity. Hence we may sum up the telescoping series resultingfrom the previous inequality to get d [Φ ( z ) , ≤ ∞ X k =1 d [0 , α k ] = ∞ X k =1 log 1 + | α k | − | α k | ≤ log Q . This, in turn, gives us the estimate1 + | Φ ( z ) | − | Φ ( z ) | ≤ ∞ Y k =1 | α k | − | α k | ≤ Q (2.16)which holds for all z ∈ D and hence also in the closure of D .
3. The inverse problem
The main purpose of [ ] was to prove the following theorem. Theorem . The locations (and multiplicities) of the zeros of the Jost func-tion, that is, the resonances, associated with the CMV equations (2.1) determineuniquely the Verblunsky coefficients, provided these satisfy | α n | ≤ η exp( − n γ ) . The strategy in [ ] was to show that the zeros of ψ determine the Weyl-Titchmarsh m -function. The m -function, in turn, determines the Verblunsky coef-ficients as was shown in [ ]. In the following we prove this fact directly, becausethe proof of our stability result relies on the details of the relationship between theresonances and the Verblunsky coefficients. Proof of Theorem 3.1.
As we know from equation (2.15), the Verblunskycoefficients are determined by the Schur functions Φ k . Of these the first one isdetermined by m while the subsequent ones are found recursively via the M¨obiustransform S .To find m we call on Schwarz’s integral formula which says that m ( z ) = 12 π Z π − π r e it + zr e it − z Re( m ( r e it )) dt (3.1)as long as | z | < r <
1. According to the first inequality in (2.16) m = (1 + z Φ ( z )) / (1 − z Φ ( z )) is uniformly bounded in D and hence has no poles in itsclosure. Therefore we may take the limit r → d ˜ µ ( t ) = Re( M (e it )) dt = dt | ψ (e it ) | so that m ( z ) = 12 π Z π − π e it + z e it − z dt | ψ (e it ) | (3.2)for | z | < ROMAN SHTERENBERG, RUDI WEIKARD, AND MAXIM ZINCHENKO
Finally we have to show that ψ is determined from its zeros (the resonances).Since it is an entire function of growth order zero Hadamard’s factorization theoremgives ψ ( z ) = ψ (0)Π( z ) whereΠ( z ) = ∞ Y n =1 (1 − z/z n )and where the z n are the zeros of ψ repeated according to their multiplicities. Weclaim that the value ψ (0) is also determined by the z n . Indeed, evaluating (3.2)at z = 0, using that m (0) = 1, gives | ψ (0) | = 12 π Z π − π dt | Π(e it ) | . (3.3)This will prove our claim if we can show that ψ (0) is positive. To this end we notethat F (0 , k ) = (1 , ⊤ is the unique solution of the Volterra-type equations (2.6)for z = 0. Thus ψ (0) = C = ∞ Y j =1 ρ − j ≥ . (3.4) (cid:3)
4. Stability
In this section we prove Theorem 1.2. Throughout the section we assume thatany sequence of Verblunsky coefficients is from the class B ( γ, η, Q ). Subsequentlywe will be using repeatedly the following elementary facts:(1) If | x | ≤ /
2, then | log(1 − x ) | ≤ | x | .(2) | e u − | ≤ | u | e | u | for all u ∈ C . Moreover, if | u | ≤ /
2, then | e u − | ≤ | u | . Π . Upper bounds on Π follow from (2.7)since Π( z ) = (1 , − F ( z,
0) by (2.10) and (3.4). Indeed, using that Q ∞ j =1 (1+ | α j | ) ≤ Q for α ∈ B ( γ, η, Q ), we findlog | Π( z ) | ≤ log( √ Q ) + η + (log 2 | z | ) γ/ ( γ − (4.1)as long as | z | ≥ M ( z )) = | ψ ( z ) | − = C − | Π( z ) | − if | z | = 1. Note also thatRe( M ( z )) = 1 − | z Φ ( z ) | | − z Φ ( z ) | ≤ | Φ ( z ) | − | Φ ( z ) | . Combining these facts with (3.4) and (2.16) gives | Π( z ) | − ≤ C ∞ Y n =1 | α n | − | α n | = ∞ Y n =1 − | α n | ) ≤ Q . (4.2) TABILITY FOR THE INVERSE RESONANCE PROBLEM FOR THE CMV OPERATOR 9
Now let N ( r ) denote the number of zerosof Π in the open disk of radius r centered at zero. We know that N ( r ) = 0 for r ≤
1. To deal with r ≥ a + b ) p ≤ (2 a ) p + (2 b ) p , which holds for a, b ≥ p ≥
1, to find N ( r ) ≤ Z err N ( t ) t dt ≤ Z er N ( t ) t dt = 12 π Z π log | Π(e r e it ) | dt ≤ A + (log r ) p p = γ/ ( γ −
1) and A is a suitable constant which depends only on Q , η , and γ . From this we obtain X | z n |≥ R | z n | = Z ∞ R dN ( t ) t = − N ( R ) R + Z ∞ R N ( t ) t dt ≤ A R + 4 p p + 1 , log R )where Γ denotes the incomplete Gamma function [ , Sect. 6.5]. In particular, weget ∞ X n =1 | z n | ≤ A + 4 p p + 1)by setting R = 1. The asymptotic behavior of the incomplete Gamma function[ , Eq. 6.5.32] shows now that X | z n |≥ R | z n | ≤ A (log R ) p R (4.3)if R ≥ A is a suitable constant (depending on Q , η , and γ ).Next we estimate how close resonances can be to the unit circle. From (4.1)(making use of the maximum principle) we know that there is a constant L depend-ing only on Q , η , and γ such that | Π( z ) | ≤ L whenever | z | ≤ e. Cauchy’s estimategives | Π ′ ( a ) | ≤ L/ (e −| a | ) for any point a with | a | < e. Let 1+ δ = ( QL +e) / ( QL +1)and z a point on the unit circle. Then | Π( tz ) | ≥ | Π( z ) | − Z t | Π ′ ( z s ) | ds ≥ Q − ( t − L e − t . Since this is positive as long as 1 ≤ t < δ we have established that there are nozeros of ψ , i.e., no resonances, in the disk | z | < δ for any operator U from theclass B ( γ, η, Q ). Π and ˘Π . We assume now that we have two CMV opera-tors U and ˘ U with Verblunsky coefficients α n and ˘ α n , respectively. More generally,any quantity associated with ˘ U will have a˘accent to distinguish it from the corre-sponding quantity associated with U . Both α and ˘ α are in B ( γ, η, Q ).Our goal is to show that the differences | α n − ˘ α n | are arbitrarily small providedthat the resonances of the associated operators U and ˘ U in a sufficiently largedisk (of radius R ) are respectively ε -close for a sufficiently small ε , i.e., to proveTheorem 1.2.We will henceforth always assume R ≥ ε ≤ δ/
2. We begin by looking atthose factors in Π and ˘Π associated with resonances which are respectively close toeach other, i.e., the resonances in a disk of radius R . Let N be their number and assume | z | = 1. Thus | ˘ z n − z n | ≤ ε and | ˘ z n − z | , | z n − z | ≥ δ > ≤ n ≤ N .Since ε/δ ≤ / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log N Y n =1 − z/ ˘ z n − z/z n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N X n =1 (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) − z z n − ˘ z n ( z n − z )˘ z n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ εδ ∞ X n =1 | ˘ z n | − . (4.4)We showed above that the sum on the right is bounded by A + Γ( p + 1) / | z n | , | ˘ z n | ≥ R ≥ | z | = 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log ∞ Y n = N +1 − z/ ˘ z n − z/z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X n = N +1 (cid:18) | ˘ z n | + 1 | z n | (cid:19) so that, with the aid of (4.3), we arrive at the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log ∞ Y n = N +1 − z/ ˘ z n − z/z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A (log R ) p R .
Combining this estimate with (4.4) and denoting by A a suitable constantdepending only on Q , η , and γ , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˘Π( z )Π( z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A (cid:18) ε + (log R ) p R (cid:19) (4.5)provided that | z | ≤ ψ and ˘ ψ . Since, by (4.5) and (4.2), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Π(e it ) | − | ˘Π(e it ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Q A (cid:18) ε + (log R ) p R (cid:19) and since C , ˘ C ≥ | C − − ˘ C − | ≤ | C − ˘ C | ≤ Q A ( ε + (log R ) p /R ) . Thus, whenever | z | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ψ ( z ) | − | ˘ ψ ( z ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A (cid:18) ε + (log R ) p R (cid:19) (4.6)where A depends only on Q , η , and γ . Suppose 1 − | w | , − | ˘ w | ≥ /Q and z, ˘ z are in the closed unit disk. Then | S ( w, z ) − S ( ˘ w, ˘ z ) | ≤ Q (4 | w − ˘ w | + 2 | z − ˘ z | ) . Since by assumption 1 − | α n | ≥ Q ∞ j =1 (1 − | α j | ) ≥ /Q , it follows that for all z onthe unit circle, | Φ k ( z ) − ˘Φ k ( z ) | ≤ Q (4 | α k − ˘ α k | + 2 | Φ k − ( z ) − ˘Φ k − ( z ) | ) . (4.7)Let k · k p denote the L p -norm on the unit circle with respect to the normalizedLebesgue measure. By Gauss’s mean value theorem | α k − ˘ α k | ≤ k Φ k − − ˘Φ k − k , k ∈ N , TABILITY FOR THE INVERSE RESONANCE PROBLEM FOR THE CMV OPERATOR 11 and hence (4.7) yields, k Φ k − ˘Φ k k ≤ Q k Φ k − − ˘Φ k − k , k ∈ N . Thus we have | α k − ˘ α k | ≤ (6 Q ) k − k Φ − ˘Φ k , k ∈ N , by induction. Since for all z on the unit circle Re M ( z ) = 1 / | ψ ( z ) | and Re ˘ M ( z ) =1 / | ˘ ψ ( z ) | are nonnegative, it follows from (2.14) that k Φ − ˘Φ k ≤ k M − ˘ M k ≤ k M − ˘ M k . The imaginary parts of M and ˘ M can be obtained from the Hilberttransform of the respective real parts. Since the Hilbert transform is unitary on thespace of square integrable functions we have k Im M − Im ˘ M k = k Re M − Re ˘ M k and hence k M − ˘ M k ≤ √ k Re M − Re ˘ M k . Thus, we get from (4.6) | α k − ˘ α k | ≤ √ Q ) k − (cid:13)(cid:13)(cid:13)(cid:13) | ψ ( z ) | − | ˘ ψ ( z ) | (cid:13)(cid:13)(cid:13)(cid:13) ≤ √ Q ) k − A (cid:18) ε + (log R ) p R (cid:19) . (4.8)Setting A = 2 √ A / (6 Q ) completes the proof of Theorem 1.2.Estimate (4.8) becomes worse with increasing k . Eventually, of course we willhave | α k − ˘ α k | ≤ η e − k γ just by using our hypothesis on super-exponential decayof the Verblunsky coefficients. Using the worst possible case and introducing yetanother approriate constant A gives us the uniform estimate | α k − ˘ α k | ≤ A (cid:18) ε + (log R ) p R (cid:19) / log(6e Q ) , k ∈ N . References [1] M. Abramowitz and I. A. Stegun,
Handbook of mathematical functions with formulas, graphs,and mathematical tables , National Bureau of Standards Applied Mathematics Series, vol. 55,For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington,D.C., 1964. MR0167642 (29
The inverse resonance problem for Jacobi opera-tors , Bull. London Math. Soc. (2005), no. 5, 727–737. MR2164835 (2006e:39032)[3] B. M. Brown, S. Naboko, and R. Weikard, The inverse resonance problem for Hermite oper-ators , Constr. Approx. (2009), no. 2, 155–174. MR2519659 (2011b:47065)[4] F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated withorthogonal polynomials on the unit circle , J. Approx. Theory (2006), no. 1-2, 172–213.MR2220038 (2007f:47027)[5] A. Iantchenko and E. Korotyaev,
Periodic Jacobi operator with finitely supported perturbationon the half-lattice , Inverse Problems (2011), no. 11, 115003, 26. MR2851909[6] A. Iantchenko and E. Korotyaev, Periodic Jacobi operator with finitely supported perturba-tions: the inverse resonance problem , J. Differential Equations (2012), no. 3, 2823–2844.MR2860642[7] A. Iantchenko and E. Korotyaev,
Resonances for periodic Jacobi operators with finitely sup-ported perturbations , J. Math. Anal. Appl. (2012), no. 2, 1239–1253.[8] E. Korotyaev,
Stability for inverse resonance problem , Int. Math. Res. Not. (2004), 3927–3936. MR2104289 (2005i:81186)[9] M. Marletta, S. Naboko, R. Shterenberg, and R. Weikard, On the inverse resonance problemfor Jacobi operators – uniqueness and stability , To appear in J. Anal. Math.[10] M. Marletta, R. Shterenberg, and R. Weikard,
On the inverse resonance problem forSchr¨odinger operators , Comm. Math. Phys. (2010), no. 2, 465–484. MR2594334(2011a:34203) [11] M. Marletta and R. Weikard,
Stability for the inverse resonance problem for a Jacobi op-erator with complex potential , Inverse Problems (2007), no. 4, 1677–1688. MR2348728(2008i:47065)[12] B. Simon, Orthogonal polynomials on the unit circle. Part 1 , American Mathematical SocietyColloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005.Classical theory. MR2105088 (2006a:42002a)[13] B. Simon,
Orthogonal polynomials on the unit circle. Part 2 , American Mathematical SocietyColloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005.Spectral theory. MR2105089 (2006a:42002b)[14] R. Weikard and M. Zinchenko,
The inverse resonance problem for CMV operators , InverseProblems (2010), no. 5, 055012, 10. MR2647154 (2011j:47098) Department of Mathematics, University of Alabama at Birmingham, Birmingham,AL 35226-1170, USA
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