Direct and inverse spectral problems for rank-one perturbations of self-adjoint operators
aa r X i v : . [ m a t h . SP ] J u l DIRECT AND INVERSE SPECTRAL PROBLEMS FORRANK-ONE PERTURBATIONS OF SELF-ADJOINTOPERATORS
OLES DOBOSEVYCH AND ROSTYSLAV HRYNIV
Abstract.
For a given self-adjoint operator A with discrete spec-trum, we completely characterize possible eigenvalues of its rank-one perturbations B and discuss the inverse problem of recon-structing B from its spectrum. Introduction
The main aim of this paper is to give a complete answer to thequestion, what spectra rank-one perturbations B = A + h· , ϕ i ψ of agiven self-adjoint operator A with simple discrete spectrum may have.There are several reasons why this question is of interest. Firstly, suchperturbations lead to the explicit formulae of perturbation theory andthus many related questions can be fully answered. Secondly, despiteits simplicity, the model offers extremely rich family of perturbed spec-tra. Namely, the main results of this paper show that, apart from theprescribed asymptotic distribution of eigenvalues, the spectrum of arank-one perturbation B of A might become arbitrary—in particular,it may get eigenvalues of arbitrarily prescribed multiplicities in an arbi-trarily prescribed finite set of complex points. In addition, we suggestan explicit method of constructing rank-one perturbations of A with agiven admissible spectrum.Similar questions in finite-dimensional case have been studied since1990-ies. In particular, Krupnik proved in [23] that, given an arbitrary n × n matrix A , the spectrum of its rank-one perturbation can becomean arbitrary complex n -tuple; that was then further specified for theclasses of Hermitian, unitary, and normal matrices. Savchenko [35]studied the changes in the Jordan structure of A under a rank-one per-turbation and found out that, generically, in each root subspace, only Date : 12 June 2020.2010
Mathematics Subject Classification.
Primary: 47A55, Secondary: 47A10,15A18, 15A60.
Key words and phrases.
Operators, rank-one perturbations, non-simple eigen-values, eigenvalue asymptotics. the longest Jordan chain splits. For low-rank perturbations, that resultwas further generalized in [36] and independently in [33]. In [17], thenumber of distinct eigenvalues of a matrix B was estimated in termsof some spectral characteristics of A and the rank of the perturbation.One should mention that earlier, H¨ormander and Melin [21] explainedsimilar effects of rank-one perturbations in an infinite-dimensional set-ting; recently, Behrndt a.o. [12] discussed possible changes to Jordanstructure of an arbitrary linear operator A in a Banach space undergeneral finite-rank perturbations.For structured matrices and matrix pencils, a detailed rank-one per-turbation theory and its application in the control theory was recentlydeveloped in a series of papers by Mehl a.o. [27–32, 38]. The resultsestablished therein include e.g. changes in the Jordan structure of A under perturbation within classes of matrices enjoying certain real orcomplex Hamiltonian symmetry [27, 29, 31], or for H -Hermitian ma-trices, with (skew-)Hermitian H , using the canonical form of the pair( B, H ) [28, 30]. Rank-one perturbations of matrix pencils and an im-portant eigenvalue placement problem were studied e.g. in [11, 19, 32],while a more general perturbation theory for structured matrices wasoutlined in the recent paper [38].The cited results are mostly essentially finite-dimensional in the sensethat their methods do not allow straightforward generalization to theinfinite-dimensional case (see, however, [14, 21]). The latter has beenstudied within the general spectral theory for bounded or unboundedoperators in infinite-dimensional Banach or Hilbert spaces [22]. For in-stance, a comprehensive spectral analysis of rank-one perturbations ofunbounded self-adjoint operators is carried out in [37], where a detailedcharacterization of discrete, absolutely continuous, and singlularly con-tinuous spectra of the perturbation B is given. A thorough overviewof the theory of Schr¨odinger operators under singular point pertur-bations (formally corresponding to additive Dirac delta-functions andtheir derivatives) is given in the monographs by Albeverio a.o. [2, 9],suggesting also comprehensive reference lists. Much attention has beenpaid to the so-called singular and super-singular rank-one or finite-rankperturbations of self-adjoint operators, where the functions ϕ and ψ belong to the scales of Hilbert spaces dom( A α ) with negative α , seee.g. [3–8, 10, 16, 18, 24–26]; in this case, a typical approach is throughthe Krein extension theory of self-adjoint operators. Rank-one andfinite-rank perturbations of self-adjoint operators in Krein spaces havebeen recently discussed in e.g. [13, 14].Despite the extensive research in the area, there seems to be no com-plete understanding what spectra rank-one perturbations of a given ANK-ONE PERTURBATIONS 3 opearator A can produce. As the earlier research demonstrates (cf. [2,9, 37]), the question is quite non-trivial even for self-adjoint perturba-tions of a self-adjoint operator A , and thus necessarily much more com-plicated for generic rank-one perturbations. In our previous work [15],we described local spectral properties of rank-one perturbations of aself-adjoint operator with discrete spectrum. Namely, it was showntherein that, as in the finite-dimensional case [23], such a perturbationcan possess eigenvalues of arbitrarily prescribed multiplicities at anyfinite set of complex numbers; cf. [20] for similar results for the classof PT -symmetric perturbations.The main aim of the present paper is to give a complete descriptionof the possible spectra of rank-one perturbations of a given self-adjointoperator A with simple discrete spectrum. More exactly, with λ n de-noting the eigenvalues of A , Theorem 3.1 states that the eigenvalues ofa rank-one perturbation B can be labelled as µ n (counting with mul-tiplicities) so that the sum of all offsets | µ n − λ n | is finite. Moreover,it turns out that every sequence µ n with this property can be a spec-trum for such a B ; Theorem 4.1 in addition suggests a method forconstructing all such rank-one perturbations B .The structure of the paper is as follows. In the next section, wecollect some basic spectral properties of the rank-one perturbations B .In Section 3, the asymptotic distribution of eigenvalues of B is stud-ied and, in particular, summability of the offsets | µ n − λ n | is proved.Sufficiency of this condition, as well as an algorithm for constructinga rank-one perturbation B with a prescribed admissible spectrum areestablished in Section 4. Finally, in Section 5, we give two examplesand discuss straightforward generalizations of the main results to widerclasses of the operators A .2. Preliminaries
In this section, we collect some properties of the rank-one pertur-bations of self-adjoint operators A acting in a fixed fixed separable(infinite-dimensional) Hilbert space H that will be required to provethe main results of this work. Throughout the whole paper, we shallassume that(A1) the operator A is self-adjoint and has simple discrete spectrum.The operator A is necessarily unbounded but it may be bounded be-low or above; without loss of generality, in this case we assume that A is bounded below (otherwise, we just replace A with − A ). Underthese assumptions, the spectrum of A consists of simple real eigenval-ues that can be listed in increasing order as λ n , n ∈ I , with I = N if O. DOBOSEVYCH AND R. HRYNIV A is bounded below and I = Z otherwise. Keeping in mind the mostimportant and interesting applications to the differential operators, wemake an additional assumption that(A2) the eigenvalues of A are separated, i.e.,(1) inf n ∈ I | λ n +1 − λ n | =: d > . The operator B is a rank-one perturbation of the operator A , i.e.,(2) B = A + h· , ϕ i ψ with fixed non-zero vectors ϕ and ψ in H and with h · , · i denoting ascalar product in H . Clearly, B is well defined and closed on its naturaldomain dom( B ) equal to dom( A ). Next, for λ ∈ ρ ( A ), we introducethe characteristic function (3) F ( λ ) := h ( A − λ ) − ψ, ϕ i + 1and denote by N F the set of zeros of F . This function appears inthe Krein resolvent formula for B [9, 15], and its zeros characterise thespectrum of B .To be more specific, we denote by v n a normalized eigenvector of A corresponding to the eigenvalue λ n ; then the set { v n } n ∈ I forms an or-thonormal basis of H , and we let a n and b n be the corresponding Fouriercoefficients of the vectors ϕ and ψ , so that ϕ = X k ∈ I a k v k , ψ = X k ∈ I b k v k . Now we set I := { n ∈ I | a n b n = 0 } , I := { n ∈ I | a n b n = 0 } and σ j ( A ) := { λ n | n ∈ I j } ; then σ ( A ) = σ ( B ) := σ ( A ) ∩ σ ( B ) isthe common part of the spectra of A and B , while the spectrum of B in C \ σ ( A ) coincides with the set of zeros of F .In fact, the function F also characterizes eigenvalue multiplicities ofthe operator B . We recall that the geometric multiplicity of an eigen-value λ of B is the dimension of the null-space of the operator B − λ ,while its algebraic multiplicity is the dimension of the correspondingroot subspace, i.e., of the set of all y ∈ dom( B ) such that ( B − λ ) k y = 0for some k ∈ N . Next, by the spectral theorem for A , the characteristicfunction F of (3) can be written as (4) F ( z ) = X k ∈ I a k b k λ k − z + 1 In what follows, the summations and products over the index sets that are notbounded from below or above are understood in the principal value sense
ANK-ONE PERTURBATIONS 5 and thus can be analytically extended to σ ( A ); we keep the notation F for this extension.As proved in [15], the geometric multiplicity of every eigenvalue µ of B is at most 2; multiplicity 2 is only possible when µ ∈ σ ( A ), i.e., µ = λ n for some n ∈ I and, in addition, a n = b n = F ( λ n ) = 0.We also observe that when a n = b n = 0, then the subspace ls { v n } isinvariant under both B and B ∗ and thus is reducing for B . Denotingby H the closed linear span of all such subspaces, we conclude that H and H ⊖ H are reducing for B and the operators A and B coincideon H . For that reason, only the part of B in H ⊖ H is of interest,and, without loss of generality, we may assume that H = { } .Under such an assumption, every eigenvalue µ of B is geometricallysimple. One of the main results of [15] claims that the algebraic mul-tiplicity m of an eigenvalue µ of B coincides with the multiplicity l of z = µ as a zero of F if µ σ ( A ) and is equal to l + 1 otherwise. Inother words, the common spectrum σ ( A ) = σ ( B ) of A and B and thezeros of F completely characterize the spectrum of B , counting withmultiplicities. This allows us to reduce the study of the eigenvalue dis-tribution for the perturbation B to the study of zero distribution ofthe characteristic function F .3. Eigenvalue distribution of the operator B In this section, we shall discuss eigenvalue distribution of the rank-one perturbation B of A given by (2). As explained in the previoussection, the spectrum of B consists of two parts: σ ( B ) = σ ( A ) ∩ σ ( B ),which is the common part of the spectra of A and B , and σ ( B ), whichis the set of zeros of the characteristic function F ( z ) = X n ∈ I a n b n λ n − z + 1in the domain C \ σ ( A ); moreover, the algebraic multiplicity of aneigenvalue µ ∈ σ ( B ) is determined by its multiplicity as a zero of thecharacteristic function F .The main result of this section is given by the following theorem. Theorem 3.1.
The eigenvalues of the operator B can be labelled as µ n , n ∈ I , in such a way that the series (5) X n ∈ I | µ n − λ n | converges. In particular, all but finitely many eigenvalues of B aresimple. O. DOBOSEVYCH AND R. HRYNIV
First we shall show that large enough elements of σ ( B ) are locatednear σ ( A ), which will enable their proper enumeration. To begin with,for k ∈ I we define the functions G k and H k by the formulas G k ( z ) = a k b k λ k − z + 1 , H k ( z ) = X (1) | n |≤ k a n b n λ n − z + 1and introduce the sets Q k := { z ∈ C | Re( z ) , Im( z ) ∈ [ λ −| k | − d , λ | k | + d ] } ,R k := { z ∈ C | | z − λ k | < d } , where we replace λ −| k | with − λ | k | if I = N . Due to the assumption (A2)the sets R k are pairwise disjoint and also R k ∩ Q n = ∅ if | k | > | n | . Lemma 3.2.
For every ε > there exist integers K ε > and K ′ ε > K ε such that the following holds: (a) for every k with | k | > K ε and every z ∈ R k = ∂R k ∪ R k (6) X (1) | n | >K ε n = k (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) < εd ;(b) for every z ∈ C \ Q K ′ ε (7) X (1) | n |≤ K ε (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) < ε. Proof.
The sequences ( a n ) n ∈ I and ( b n ) n ∈ I of the Fourier coefficients ofthe vectors φ and ψ are square summable, so that, by the Cauchy–Bunyakowsky–Schwarz inequality, X n ∈ I | a n b n | < ∞ . Therefore, for every ε > K ε such that X (1) | n | >K ε | a n b n | < ε. Take a k satisfying | k | > K ε ; then by virtue of Assumption (A2) forevery z ∈ R k and every n = k we get | λ n − z | ≥ d , and therefore (6)holds.For part (b), note that | λ n − z | > ( K ′ ε − K ε ) d if | n | ≤ K ε and z ∈ C \ Q k with | k | ≥ K ′ ε > K ε ; therefore, by choosing K ′ ε largeenough, we arrive at (7). (cid:3) Throughout the paper, the symbol P (1) will denote summation over the indexset I ANK-ONE PERTURBATIONS 7
Corollary 3.3.
Take ε := d/ (2 + d ) and K ′ ε as in the above lemma;then σ ( B ) ⊂ Q K ′ ε ∪ (cid:16)[ n ∈ I R n (cid:17) . Indeed, it suffices to note that if z is outside Q K ′ ε and every R n , n ∈ I , then | λ n − z | ≥ d/ , so that X (1) | n | >K ε (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) < εd , which together with part (b) of that lemma shows that | F ( z ) | ≥ − X (1) | n | >K ε (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) > − ε (1 + 2 /d ) = 0 . Lemma 3.4.
There exists
K > such that for all k ∈ I with | k | > K the following holds: (a) the function F has exactly one zero in R k ; (b) the functions H k and F have the same number of zeros in Q k .Proof. Fix an ε ∈ (0 , d/
2) such that ε (cid:16) d (cid:17) < K = K ′ ε of Lemma 3.2.If k satisfies | k | > K , then by Lemma 3.2 for every z ∈ ∂R k we get | F ( z ) − G k ( z ) | ≤ X (1) | n |≤ K ε (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) + X (1) | n | >K ε n = k (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) < ε + 2 εd . On the other hand, | a k b k | < ε if k ∈ I satisfies | k | > K > K ε , andthen(8) | G k ( z ) | ≥ − (cid:12)(cid:12)(cid:12)(cid:12) a k b k λ − λ k (cid:12)(cid:12)(cid:12)(cid:12) > − εd for all z ∈ ∂R k . By the choice of ε we conclude that then(9) | G k ( z ) | > | F ( z ) − G k ( z ) | for all such z . As the functions G k and F both have the same number ofpoles in R k (namely, a simple pole at λ k ), by estimate (9) and Rouche’stheorem they have the same number of zeros in the set R k . By virtue ofinequality (8), the unique zero z = λ k + a k b k of the function G k belongsto the circle R k for all k ∈ I with | k | > K , and thus the function F has exactly one zero in R k for such k as well. This completes the proofof part (a). O. DOBOSEVYCH AND R. HRYNIV
Next, by the definition of the set Q k , it holds that | λ n − z | ≥ d if z ∈ ∂Q k and | n | > | k | . By the choice of the number K ε , we find that | F ( z ) − H k ( z ) | ≤ X (1) | n | > | k | (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) < εd and(10) X (1) K ε < | n |≤| k | (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) < εd if | k | > K ε and z ∈ ∂Q k . Also, by part (b) of Lemma 3.2 we have(11) X (1) | n |≤ K ε (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) < ε as soon as | k | > K and z ∈ C \ Q k . Combining estimates (10) and(11), we conclude that(12) | H k ( z ) | ≥ − X (1) | n |≤ k (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) > − ε − εd for all k with | k | > K and all z ∈ C \ Q k .It follows that for k with | k | > K and for all z ∈ ∂Q k | H k ( z ) | > | F ( z ) − H k ( z ) | . Since the functions H k and F have the same poles in Q k (namely, simplepoles λ n for n ∈ I with | n | ≤ k ), we conclude by Rouche’s theoremthat they have the same number of zeros in Q k for all k > K . Theproof is complete. (cid:3) Remark 3.5.
Take k larger than K of the above lemma and denote by N k the cardinality of the set σ ( A ) ∩ Q k . The function H k is a ratio oftwo polynomials of degree N k and due to (12) all its zeros are in Q k .Therefore, the function F has precisely N k zeros in Q k counting withmultiplicities. Corollary 3.6.
The zeros of F in C \ σ ( A ) can be labelled (countingwith multiplicities) as µ k with k ∈ I in such a way that | µ k − λ k | < d for all k ∈ I with | k | > K . Recalling the results of the previous section on relation between theeigenvalues of B and zeros of the function F in C \ σ ( A ), we arrive atthe following conclusion. Corollary 3.7.
Eigenvalues of the operator B can be labelled (countingwith mulitplicities) as µ k with k ∈ I in such a way that | µ k − λ k | < d when | k | > K , K being the constant of Lemma 3.4. ANK-ONE PERTURBATIONS 9
Proof of Theorem 3.1.
We fix an enumeration of µ k as in Corollary 3.7.Then µ k = λ k for all k ∈ I with sufficiently large | k | , whence it sufficesto prove that the series X n ∈ I | µ n − λ n | is convergent.We take ε and K as in Lemma 3.4; then, according to Corollary 3.7,for every k ∈ I with | k | > K the eigenvalue µ k ∈ R k is a zero of F , sothat F ( µ k ) = X (1) | n |≤ K ε a n b n µ k − λ n + X (1) | n | >K ε n = k a n b n µ k − λ n + a k b k µ k − λ k + 1 = 0and (cid:12)(cid:12)(cid:12)(cid:12) a k b k µ k − λ k (cid:12)(cid:12)(cid:12)(cid:12) > − X (1) | n |≤ K ε (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − µ k (cid:12)(cid:12)(cid:12)(cid:12) − X (1) | n | >K ε n = k (cid:12)(cid:12)(cid:12)(cid:12) a n b n λ n − µ k (cid:12)(cid:12)(cid:12)(cid:12) . By virtue of Lemma 3.4 we conclude that (cid:12)(cid:12)(cid:12)(cid:12) a k b k µ k − λ k (cid:12)(cid:12)(cid:12)(cid:12) > − ε − εd for k ∈ I with | k | > K . Since 1 − ε − εd > εd , we find that(13) | µ k − λ k | < d ε | a k b k | for k ∈ I with | k | > K . As the series P n ∈ I | a n b n | is convergent, thesame is true of the series P n ∈ I | µ n − λ n | , and the proof is complete. (cid:3) Inverse spectral problem
The purpose of this section is to study the inverse spectral prob-lem, namely, the problem of reconstructing the operator B from itsspectrum ( µ n ) n ∈ I assuming that the operator A is known.More generally, let the operator A satisfy assumptions (A1) and(A2), i.e., is self-adjoint and has a simple discrete spectrum ( λ n ) n ∈ I that is d -separated as in (1). Our aim is to find necessary and sufficientconditions that another sequence ( ν n ) n ∈ I of complex numbers mustsatisfy so that it could be a spectrum (counting with multiplicities) ofan operator B of the form (2). Also, we want to suggest an algorithm ofconstructing the operator B (i.e., the function ϕ and ψ ) and investigatethe uniqueness question. The latter question can be answered straight ahead. Indeed, if theinverse problem for a sequence ( ν n ) n ∈ I has a solution, then it has manysolutions. In fact, if B j = A + h· , ϕ j i ψ j , j = 1 , ϕ , ϕ , ψ , ψ satisfy h ϕ , v n ih ψ , v n i = h ϕ , v n ih ψ , v n i , n ∈ N , then the spectra of B and B coincide counting with multiplicities.Therefore, in the inverse problem one can only restore the products a n b n of the Fourier coefficients of the functions ϕ and ψ , which are theresidues of the function − F of (4).The main result of this section is given by the following theorem. Theorem 4.1.
Assume that a sequence ν of complex numbers can beenumerated as ν n , n ∈ I , in such a way that the series (14) X n ∈ I | ν n − λ n | converges. Then there exist vectors ϕ, ψ ∈ H such that the spectrumof B coincides with ν counting with multiplicities. Let us denote by I the set of indices n ∈ I for which λ n appearsin ν and set Λ := { λ n | n ∈ I } . Convergence of the series (14) impliesthat for every ε ∈ (0 , d/
2) there exists a
K > | ν n − λ n | < ε for all n ∈ I with | n | > K . Therefore, if n ∈ I and | n | > K , then ν n = λ n , and without loss of generality we may assume that ν n = λ n for all n ∈ I .We also set I := I \ I , Λ := { λ n | n ∈ I } , and introduce thefunction(15) ˜ F ( z ) := Y n ∈ I ν n − zλ n − z . To show that ˜ F is well defined, we take an arbitrary ε ∈ (0 , d/
2) andset R n ( ε ) := { z ∈ C | | z − λ n | < ε } , R ( ε ) := C \ (cid:0) ∪ n ∈ I R n ( ε ) (cid:1) . Then we have the following
Lemma 4.2.
For each ε ∈ (0 , d/ , the product in (15) convergesuniformly in R ( ε ) .Proof. It is enough to show that the series X n ∈ I log (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) ν n − λ n λ n − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ANK-ONE PERTURBATIONS 11 converges uniformly on the same set. However, for z ∈ R ( ε ) and n ∈ I we get the estimate(16) log (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) ν n − λ n λ n − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ν n − λ n λ n − z (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ν n − λ n | ε , which in view of the convergence of the series (14) and the WeierstrassM-test finishes the proof. (cid:3) The Weierstrass M -test used in the above proof also justifies passageto the limit lim u → + ∞ X n ∈ I (cid:12)(cid:12)(cid:12)(cid:12) ν n − λ n λ n − iu (cid:12)(cid:12)(cid:12)(cid:12) = 0;as a result, we get Corollary 4.3.
There exists the limit lim u → + ∞ ˜ F ( iu ) = 1 . The function ˜ F is meromorphic in C , and its residue at the point λ n ∈ Λ is(17) − c n = lim z → λ n ( z − λ n ) ˜ F ( z ) = ( λ n − ν n ) Y m ∈ I m = n ν m − λ n λ m − λ n . Lemma 4.4.
The series (18) X n ∈ I | c n | converges.Proof. In view of (17), convergence of series (18) follows from conver-gence of the series X n ∈ I | λ n − ν n | Y m ∈ I m = n (cid:12)(cid:12)(cid:12)(cid:12) ν m − λ n λ m − λ n (cid:12)(cid:12)(cid:12)(cid:12) , and to establish the latter it is enough to show that the sequence(19) Y m ∈ I m = n (cid:12)(cid:12)(cid:12)(cid:12) ν m − λ n λ m − λ n (cid:12)(cid:12)(cid:12)(cid:12) is bounded in n ∈ I . Applying the same reasoning as in the proof of Lemma 4.2, we con-clude that the sum of the series X (1) m = n log (cid:12)(cid:12)(cid:12)(cid:12) ν m − λ n λ m − λ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ X (1) m = n log (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) ν m − λ m λ m − λ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ X (1) m = n (cid:12)(cid:12)(cid:12)(cid:12) ν m − λ m λ m − λ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ d X m ∈ I | ν m − λ m | has an n -independent bound, which implies that the sequence (19) isuniformly bounded. The proof is complete. (cid:3) In view of the above lemma, the series X n ∈ I c n λ n − z converges uniformly in R ( ε ) for every ε ∈ (0 , d/ F ( z ) := 1 + X n ∈ I c n λ n − z is well defined and analytic in the set C \ Λ and has simple poles atthe points z ∈ Λ . The Lebesgue dominated convergence theorem alsoimplies that lim u → + ∞ F ( iu ) = 1 . Lemma 4.5.
The function F − ˜ F is equal to zero identically in C .Proof. We set G := F − ˜ F ; then the function G is meromorphic in C with possible single poles at the points Λ . However, as the residua of F and ˜ F at each point z ∈ Λ coincide by construction, we concludethat the function G has removable singularities at the points z ∈ Λ and thus is entire. We next show that G is uniformly bounded over C and thus is constant by the Liouville theorem; aslim u → + ∞ G ( iu ) = lim u → + ∞ F ( iu ) − lim u → + ∞ ˜ F ( iu ) = 0 , this constant is zero, and thus the proof will be complete.For large enough k , we denote by Q k the rectangular bounded by thelines Im z = ± λ k , Re z = λ −| k | − d/
2, and Re z = λ k + d/ I = N ,then we replace λ − k with − λ k ). Observe that for every n ∈ I and z ∈ ∂Q k we have | z − λ n | ≥ d/
2; as a result, we conclude thatsup z ∈ ∂Q k | F ( z ) | ≤ d X n ∈ I | c n | := C. ANK-ONE PERTURBATIONS 13
Next, we note that for ε ∈ (0 , d/
2) the boundary ∂Q k of Q k lies inthe set R ( ε ). As in the proof of Lemma 4.2, we can derive the bound(cf. (16)) sup z ∈ ∂Q k | ˜ F ( z ) | ≤ exp n ε X n ∈ I | ν n − λ n | o := ˜ C. Since the function G is entire, it follows from the maximum modulusprinciple that | G ( z ) | ≤ C + ˜ C inside every set Q n and thus for all z ∈ C . Therefore, the function G is bounded; as explained at the beginning of the proof, this implies therequired results. (cid:3) Proof of Theorem 4.1.
Given any sequence ν of complex numbers sat-isfying the assumption of the theorem, we construct the meromorphicfunction ˜ F via (15). Next, calculate the residua − c n of ˜ F at the points λ n ∈ Λ via (17) and define the sequences(20) a n := p | c n | , b n := p | c n | e i arg c n , n ∈ I , and a n := 1 / (1 + | n | ) , b n = 0 , n ∈ I . Since the sequence ( c n ) n ∈ I is summable by Lemma 4.4, it followsthat the sequences ( a n ) n ∈ I and ( b n ) n ∈ I belong to ℓ ( I ). Therefore,there exist functions ϕ and ψ in the Hilbert space H whose Fouriercoefficients in the basis v n are equal to a n and b n , respectively.We now consider the operator B of the form (2) with the functions ϕ and ψ just introduced and conclude by virtue of Lemma 4.5 that thecorresponding meromorphic function F of (4) coincides with ˜ F . There-fore, zeros of F are precisely the elements of the subsequence ν :=( ν n ) n ∈ I , both counting multiplicity; namely, if a number ν occurs k times in ν , it is a zero of F of multiplicity k . The analysis of thepaper [15] summarized in Section 2 shows that each element ν of ν isan eigenvalue of B and its multiplicity is equal to the number of times ν is repeated in the sequence ν . The proof is complete. (cid:3) The above proof also gives an algorithm of constructing an opera-tor B for any sequence ν of complex numbers satisfying (14). Namely,given such a sequence ν , we(1) construct the product ˜ F of (15);(2) then calculate the residua − c n of ˜ F at the points λ n ;(3) construct the Fourier coefficients a n and b n of ϕ and ψ via (20). As was noted at the beginning of this section, there are infinitely manysuch operators; all of them are fixed by the condition a n b n = c n on theFourier coefficient a n and b n of the functions ϕ and ψ .5. Examples and discussions
We give here two examples illustrating that the results of the pa-per are in a sense optimal. For simplicity, we take the unperturbedoperator A to be defined in the Hilbert space L (0 , π ) via A = 1 i ddx subject to the periodic boundary condition y (0) = y (2 π ). The spec-trum of A coincides with the set Z , and a normalized eigenfunction v n corresponding to the eigenvalue λ n := n is equal to e inx / √ π . There-fore, the characteristic function of a generic rank-one perturbation B of (2) has the form F ( z ) = X n ∈ Z c n n − z + 1 , where c n := a n b n is determined via the Fourier coefficients a n and b n of the functions ϕ and ψ . Example 5.1.
Our first example shows that convergence of the se-ries (5) is not guaranteed if the functions ϕ and ψ do not belong to L (0 , π ). Namely, we take a n = a − n = n − / and b n = − b − n = n − / for n ∈ N and a = b = 0; thus c n = n − for n = 0. To study theasymptotics of the corresponding eigenvalues µ n of the operator B , werecall the equality [1, Ch. 5.2] X n ∈ Z n =0 n ( n − z ) = 1 z − πz cot πz ;thus F ( z ) = X n ∈ Z n =0 n ( n − z ) + 1 = z + 1 z − πz cot πz. It follows that µ n are zeros of the equationtan πz = πzz + 1and thus µ n = λ n + ε n with ε n → | n | → ∞ ; the relation µ n ε n ( µ n + 1) = tan πε n πε n → ANK-ONE PERTURBATIONS 15 as | n | → ∞ now implies that ε n µ n →
1, and thus ε n = n − (1 + o (1))as | n | → ∞ . As a result, the series (5) diverges. Example 5.2.
Consider the rank-one perturbation B of A as in (2)with ϕ and ψ given by their Fourier coefficients a = b = 0 and a n = a − n = n − β and b n = b − n = n − β for n ∈ N , with β >
1. We observethat the functions ϕ and ψ can be found explicitly via the fractionalderivatives, cf. [39]. The corresponding characteristic function F isequal to F ( z ) = 1 + X n =0 | n | − β n − z , and can be also represented as a product F ( z ) = Y n ∈ Z n =0 µ n − zn − z . The proof of Theorem 3.1 (see (13)) shows that µ n − n = O ( | n | − β ) as | n | → ∞ . The residue of F at the point z = n is equal to −| n | − β ; onthe other hand, it can be calculated as (cf. (17))res z = n F ( z ) = ( n − µ n ) Y m ∈ Z m = n µ m − nm − n . The infinite products in the above formula have been shown in theproof of Lemma 4.4 to be uniformly bounded in n (cf. the reasoningfollowing formula (19)). Therefore, we conclude that | n | − β ≤ C | µ n − n | , for a constant C independent of n , so that | µ n − n | ≍ | n | − β . Remark 5.3.
The same arguments lead to conclusion that, for a genericrank-one perturbation, | µ n − λ n | ≍ | a n b n | as | n | → ∞ . This allows usto control the decay of the offsets | µ n − λ n | through the products | a n b n | of the Fourier coefficients of ϕ and ψ and vice versa. Fix an arbitrary function ϕ ∈ H and let a n be its Fourier coefficientsin the orthonormal basis ( v n ) n ∈ I of the eigenfunctions v n of A . Denoteby ℓ ( ϕ ) the subspace of ℓ ( I ) consisting of all sequences c = ( c n ) n ∈ I of the form c n = x n a n with ( x n ) n ∈ I ∈ ℓ ( I ). The above analysis leadto the following uniqueness result: Corollary 5.4.
Given ϕ ∈ H , for every sequence ε = ( ε n ) n ∈ I ∈ ℓ ( ϕ ) there exists a function ψ ∈ H such that the rank-one perturbation B of the operator A given by (2) has eigenvalues µ n := λ n + ε n , n ∈ I ,counting with multiplicities.Such ψ is unique if and only if none of a n vanishes; each n ∈ I such that a n = 0 leaves the corresponding Fourier coefficient b n unde-termined and thus increases by one the degree of freedom of the set ofall such ψ .The roles of ϕ and ψ can be interchanged. We conclude the paper with some comments on the results obtained.Most of the analysis of [15] and of this paper has straightforward gen-eralization to the case of a normal operator A . The most crucial prop-erties and facts used are(a) the spectrum of A is simple and separated;(b) the eigenvectors form an orthonormal basis (or even a Rieszbasis) of H ;(c) the spectral theorem allowing to represent the characteristicfunction F of a rank-one perturbation B in the form (4).Some care should be given to properly choose the regions Q k in Sec-tion 3 and 4, but otherwise the arguments remain valid and establishTheorems 3.1 and 4.1, i.e., justify the possibility to enumerate the spec-trum of B so that series (5) converges and, for every sequence ( ν n ) n ∈ I satisfying (14), to construct a rank-one perturbation B of A whosespectrum is given by that sequence counting with multiplicities.In the special case of a self-adjoint rank-one perturbation (2) with ψ = αϕ and α ∈ R , the resulting spectrum of B is simple outside σ ( A ),of geometric multiplicity at most 2 at the points of σ ( A ), and theeigenvalues σ ( A ) and σ ( B ) strictly interlace, i.e., between every twoconsecutive eigenvalues from σ ( A ) there is a unique eigenvalue from σ ( B ) and, vice versa, between every two consecutive eigenvalues from σ ( B ) there is a unique eigenvalue from σ ( A ). This interlacing prop-erty follows from the minmax principle [34]; moreover, for α > λ n < µ n for all n ∈ I ; the signs are reversed if α <
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