aa r X i v : . [ m a t h . SP ] M a r ON TWO-SPECTRA INVERSE PROBLEMS
NAMIG J. GULIYEV
Abstract.
We consider a two-spectra inverse problem for the one-dimensionalSchr¨odinger equation with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter and provide a complete so-lution of this problem.
Contents
1. Introduction and main result 12. Preliminaries 33. Properties of two problems with a common boundary condition 64. Inverse problem 7Appendix A. Auxiliary results 9References 111.
Introduction and main result
The study of two-spectra inverse problems was initiated by Borg [4], who provedthat the potential q of the one-dimensional Schr¨odinger equation − y ′′ ( x ) + q ( x ) y ( x ) = λy ( x ) (1.1)is uniquely determined by the spectra of the boundary value problems generatedby this equation and the boundary conditions y ′ (0) = h y (0), y ′ ( π ) = Hy ( π ) and y ′ (0) = h y (0), y ′ ( π ) = Hy ( π ) respectively (with h = h ). Subsequent develop-ments by Marchenko [20], Krein [15], Levitan and Gasymov [17], [18] and othersshowed that not only the potential q but also the boundary coefficients h , h and H are uniquely determined by these spectra, and that any two interlacing sequencessatisfying certain asymptotic conditions are indeed the spectra of boundary valueproblems of the above form (see also [7], [21]). These results were relatively recentlygeneralized to problems with distributional potentials [6], [13], [23].In this paper we are interested in two-spectra inverse problems for boundaryvalue problems with boundary conditions dependent on the eigenvalue parameter.Such problems have also been considered in the literature. Some uniqueness resultswere obtained in [1], [2], [3], [8]. The papers [5], [19] contain some existence resultsfor problems with one eigenvalue-parameter-dependent boundary condition. In thecase when only one of the boundary conditions depends linearly on the eigenvalueparameter, necessary and sufficient conditions for solvability of the two-spectra Mathematics Subject Classification.
Key words and phrases. two-spectra inverse problem, one-dimensional Schr¨odinger equation,boundary conditions dependent on the eigenvalue parameter. inverse problem were found in [16], [9]. For problems with coupled boundary con-ditions dependent on the eigenvalue parameter, see [14] and the references therein.We consider two-spectra inverse problems for boundary value problems generatedby the equation (1.1) together with boundary conditions of the form y ′ (0) y (0) = − f ( λ ) , y ′ ( π ) y ( π ) = F ( λ ) , (1.2)where q ∈ L (0 , π ) is real-valued and f ( λ ) = h λ + h + d X k =1 δ k h k − λ , F ( λ ) = H λ + H + D X k =1 ∆ k H k − λ (1.3)are rational Herglotz–Nevanlinna functions with real coefficients, i.e., h , H ≥ δ k , ∆ k > h < . . . < h d , H < . . . < H D . Using Darboux-type transformationsbetween such boundary value problems, we recently obtained in [11] various directand inverse spectral results for boundary value problems of the form (1.1), (1.2).But these transformations are not applicable to two-spectra inverse problems be-cause a pair of boundary value problems with a common boundary condition istransformed to a pair of boundary value problems with no common boundary con-ditions. Therefore we first reduce our two-spectra problem to an inverse problemsolved in [11] and then completely solve the two-spectra problem.We denote the boundary value problem (1.1)-(1.2) by P ( q, f, F ), and assumethat α = 0 and ind f ≥ P ( q, f, F ) and P ( q, f + α, F ) aredifferent. We use the notation x n = y n + ℓ (cid:18) n (cid:19) to mean P ∞ n =0 | n ( x n − y n ) | < ∞ . We also assume that no eigenvalue of P ( q, f, F )is a pole of f or, which is the same, the spectra of the problems P ( q, f, F ) and P ( q, f + α, F ) do not intersect. It turns out that the problems P ( q, f, F ) and P ( q, f + α, F ) are not, in general, uniquely determined by their spectra. However,we are able to describe all pairs of problems with the given two spectra. Our mainresult reads as follows. Theorem 1.1.
Two sequences { λ n } n ≥ and { µ n } n ≥ are the eigenvalues of a pairof problems of the form P ( q, f, F ) and P ( q, f + α, F ) if and only if they interlaceand satisfy asymptotics of the form p λ n = n − L + σπn + ℓ (cid:18) n (cid:19) , p λ n − p µ n = ( n − L ) − r − (cid:18) ν + ℓ (cid:18) n (cid:19)(cid:19) for some integer or half-integer L ≥ − / , σ ∈ R , ν ∈ R \ { } and r ∈ { , } ,with the exception of the case when L = − / and r = 1 . Moreover, there is aone-to-one correspondence between such problems and sets of nonnegative integersof cardinality not exceeding L + (1 − r ) / . In particular, the proof of this theorem (Section 4) also yields the followinguniqueness result.
Corollary 1.2.
The problems P ( q, f, F ) and P ( q, f + α, F ) are uniquely determinedby their spectra and the poles of f . N TWO-SPECTRA INVERSE PROBLEMS 3
So in a sense, the amount of spectral data required for the unique determinationin our case (two spectra and a finite number of indices) is between that of theclassical case (two spectra only) and of problems with coupled boundary conditions(two spectra and an infinite sequence of signs); for the latter case see, e.g., [12] andthe references therein.The paper is organized as follows. In Section 2 we introduce the necessarynotation and obtain some useful identities. In Section 3 we find some conditions forthe eigenvalues of the problems P ( q, f, F ) and P ( q, f + α, F ). Section 4 is devotedto the proof of our main result, Theorem 1.1. Finally, in the appendix we provetwo auxiliary lemmas used in the main text.2. Preliminaries
First we introduce some further notation. We assign to each function f of theform (1.3) two polynomials f ↑ and f ↓ by writing this function as f ( λ ) = f ↑ ( λ ) f ↓ ( λ ) , where f ↓ ( λ ) := h ′ d Y k =1 ( h k − λ ) , h ′ := ( /h , h > , , h = 0 . We define the index of f as ind f := deg f ↑ + deg f ↓ . If f = ∞ then we just set f ↑ ( λ ) := − , f ↓ ( λ ) := 0 , ind f := − . It can easily be verified that each nonconstant function f of the form (1.3) isstrictly increasing on any interval not containing any of its poles, and f ( λ ) → ±∞ (respectively, f ( λ ) → h ) as λ → ±∞ if its index is odd (respectively, even).Let ϕ ( x, λ ), ψ ( x, λ ) and χ ( x, λ ) be the solutions of (1.1) satisfying the initialconditions ϕ (0 , λ ) = f ↓ ( λ ) , ψ (0 , λ ) = f ↓ ( λ ) , χ ( π, λ ) = F ↓ ( λ ) ,ϕ ′ (0 , λ ) = − f ↑ ( λ ) , ψ ′ (0 , λ ) = − f ↑ ( λ ) − αf ↓ ( λ ) , χ ′ ( π, λ ) = F ↑ ( λ ) . (2.1)Then the eigenvalues of the boundary value problems P ( q, f, F ) and P ( q, f + α, F )coincide with the zeros of (their characteristic functions )Φ( λ ) := F ↑ ( λ ) ϕ ( π, λ ) − F ↓ ( λ ) ϕ ′ ( π, λ ) = f ↓ ( λ ) χ ′ (0 , λ ) + f ↑ ( λ ) χ (0 , λ )andΨ( λ ) := F ↑ ( λ ) ψ ( π, λ ) − F ↓ ( λ ) ψ ′ ( π, λ ) = f ↓ ( λ ) χ ′ (0 , λ ) + ( f ↑ ( λ ) + αf ↓ ( λ )) χ (0 , λ )respectively. These eigenvalues have the asymptotics (see [11, Theorem 4.1] fordetails) p λ n = n − ind f + ind F σπn + ℓ (cid:18) n (cid:19) (2.2)and √ µ n = n − ind f + ind F σ ′ πn + ℓ (cid:18) n (cid:19) NAMIG J. GULIYEV with σ − σ ′ = ( α, ind f is even0 , ind f is odd.In the next section we will obtain more refined asymptotics for the difference of thesquare roots of the eigenvalues λ n and µ n .Since for each eigenvalue λ n of P ( q, f, F ) the solutions ϕ ( x, λ n ) and χ ( x, λ n ) arelinearly dependent, there exists a unique number β n = 0 such that χ ( x, λ n ) = β n ϕ ( x, λ n ) . (2.3)The norming constants of the problem P ( q, f, F ) are defined as γ n := Z π ϕ ( x, λ n ) d x + f ′ ( λ n ) f ↓ ( λ n ) + F ′↑ ( λ n ) F ↓ ( λ n ) − F ↑ ( λ n ) F ′↓ ( λ n ) β n . They have the asymptotics ([11, Theorem 4.1]) γ n = π (cid:18) n − ind f + ind F (cid:19) f (cid:18) ℓ (cid:18) n (cid:19)(cid:19) . (2.4)The sequences { λ n } n ≥ , { β n } n ≥ and { γ n } n ≥ are related by the identity ([11,Lemma 2.1]) Φ ′ ( λ n ) = β n γ n . (2.5)In the remaining part of this section, we are going to show that the coefficientsof the polynomial f ↓ ( λ ) satisfy a nonsingular system of linear equations whosecoefficients are expressed in terms of the sequences { λ n } n ≥ and { γ n } n ≥ . Thusany polynomial whose coefficients satisfy this system must necessarily coincide with f ↓ ( λ ). We will need this result in Section 4.We start with some identities for the eigenvalues and the norming constants ofthe problem P ( q, f, F ). Such identities are characteristic to problems with boundaryconditions dependent on the eigenvalue parameter; they were used in [10] to obtainexplicit expressions for all the coefficients of the boundary conditions in the caseof linear dependence on the eigenvalue parameter (i.e., ind f = ind F = 2 in ournotation). Lemma 2.1.
The following identities hold: ∞ X n =0 λ kn f ↓ ( λ n ) γ n = 0 , k = 0 , . . . , d − . Proof.
From (2.1) and (2.3) we have f ↓ ( λ n ) = ϕ (0 , λ n ) = χ (0 , λ n ) β n . Together with (2.5) this implies (for sufficiently large N ) N X n =0 λ kn f ↓ ( λ n ) γ n = N X n =0 Res λ = λ n λ k χ (0 , λ )Φ( λ ) = 12 π i Z C N λ k χ (0 , λ )Φ( λ ) d λ, where C N is the circle of radius (cid:18) N − ind f + ind F − (cid:19)
2N TWO-SPECTRA INVERSE PROBLEMS 5 centered at the origin. Expressing χ ( x, λ ) as a linear combination of the cosine-and sine-type solutions we obtain χ ( x, λ ) = O (cid:18)(cid:12)(cid:12)(cid:12) √ λ (cid:12)(cid:12)(cid:12) ind F e | Im √ λπ | (cid:19) . On the otherhand, from (A.1) we get (see, e.g., the proof of [7, Theorem 1.1.3] for details)1Φ( λ ) = O (cid:18)(cid:12)(cid:12)(cid:12) √ λ (cid:12)(cid:12)(cid:12) − (ind f +ind F +1) e −| Im √ λπ | (cid:19) , λ ∈ [ N C N , and thus λ k χ (0 , λ )Φ( λ ) = O (cid:18) N ind f − k +1 (cid:19) , λ ∈ [ N C N with ind f − k + 1 ≥
3. Hencelim N →∞ Z C N λ k χ (0 , λ )Φ( λ ) d λ = 0 , which proves the lemma. (cid:3) Denote by p d − , . . . , p the non-leading coefficients of the polynomial f ↓ ( λ ) afterdividing it by its leading coefficient:( − d h ′ f ↓ ( λ ) = d Y k =1 ( λ − h k ) = λ d + p d − λ d − + . . . + p λ + p . It is easy to see from the asymptotics of the eigenvalues and the norming constantsthat for each k = 0, . . . , d − s k := ∞ X n =0 λ kn γ n converges absolutely. Lemma 2.1 implies the following identities between the num-bers p i and s j : d − X i =0 p i s i + k = − s d + k , k = 0 , , . . . , d − . (2.6)We consider them as a system of linear equations (with respect to the numbers p i ),the matrix of which is the following Hankel matrix: s s . . . s d − s s . . . s d ... ... . . . ... s d − s d . . . s d − The quadratic form corresponding to this matrix is positive definite: d − X i,j =0 s i + j ξ i ξ j = d − X i,j =0 ∞ X n =0 λ i + jn ξ i ξ j γ n = ∞ X n =0 d − X i,j =0 λ i + jn ξ i ξ j γ n = ∞ X n =0 γ n d − X i =0 λ in ξ i ! ≥ P d − i =0 λ in ξ i = 0 for all n , i.e. ξ = . . . = ξ d − = 0. Thusthe determinant of the above matrix is strictly positive and hence the system (2.6)has a unique solution. NAMIG J. GULIYEV Properties of two problems with a common boundary condition
We are now going to study further properties of the sequences { λ n } n ≥ and { µ n } n ≥ . We will first show that these two sequences interlace and then find morerefined asymptotics for the difference of their square roots. As we will see in thenext section, any two sequences with these two properties are indeed the eigenvaluesof a pair of boundary value problems with a common boundary condition.The function m ( λ ) := − Ψ( λ )Φ( λ )satisfies the identity m ( λ ) = m ( λ ) and is a meromorphic function with poles at λ n and zeros at µ n . For nonreal values of λ the solution y ( x, λ ) := ψ ( x, λ ) + m ( λ ) ϕ ( x, λ )satisfies the boundary condition F ↑ ( λ ) y ( π, λ ) − F ↓ ( λ ) y ′ ( π, λ ) = 0 . Using (2.1) we calculate( λ − µ ) Z π y ( x, λ ) y ( x, µ ) d x = ( y ( x, λ ) y ′ ( x, µ ) − y ′ ( x, λ ) y ( x, µ )) (cid:12)(cid:12)(cid:12)(cid:12) π = ( F ( µ ) − F ( λ )) y ( π, λ ) y ( π, µ ) + αf ↓ ( λ ) f ↓ ( µ ) ( m ( λ ) − m ( µ ))+ ( f ↓ ( λ ) f ↑ ( µ ) − f ↓ ( µ ) f ↑ ( λ )) (1 + m ( λ )) (1 + m ( µ )) . For µ = λ this yields α Im m ( λ )Im λ = 1 | f ↓ ( λ ) | Z π | y ( x, λ ) | d x + (cid:12)(cid:12)(cid:12)(cid:12) y ( π, λ ) f ↓ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) Im F ( λ )Im λ + | m ( λ ) | Im f ( λ )Im λ > . Thus αm ( λ ) is a Herglotz–Nevanlinna function, and hence its zeros µ n and poles λ n interlace.Using (2.1), (2.3) and the constancy of the Wronskian we obtainΨ( λ n ) = F ↑ ( λ n ) ψ ( π, λ n ) − F ↓ ( λ n ) ψ ′ ( π, λ n )= β n ( ϕ ′ ( π, λ n ) ψ ( π, λ n ) − ϕ ( π, λ n ) ψ ′ ( π, λ n ))= β n ( ϕ ′ (0 , λ n ) ψ (0 , λ n ) − ϕ (0 , λ n ) ψ ′ (0 , λ n )) = αβ n f ↓ ( λ n ) . Together with (2.5) this implies γ n = αf ↓ ( λ n )Φ ′ ( λ n )Ψ( λ n ) . We will need this formula in the next section in order to transform our two-spectrainverse problem to an inverse problem solved in [11], but for now we will use it toobtain more refined asymptotics for the difference √ λ n − √ µ n . The mean valuetheorem yieldsΨ( λ n ) = Ψ( λ n ) − Ψ( µ n ) = (cid:16)p λ n − p µ n (cid:17) (cid:16)p λ n + p µ n (cid:17) Ψ ′ ( ζ n ) (3.1) N TWO-SPECTRA INVERSE PROBLEMS 7 for ζ n ∈ [ λ n , µ n ] with √ ζ n = n − (ind f + ind F ) / O (cid:0) n (cid:1) . Thus p λ n − p µ n = αf ↓ ( λ n )Φ ′ ( λ n ) (cid:0) √ λ n + √ µ n (cid:1) γ n Ψ ′ ( ζ n ) . Applying Lemma A.1 to the problems P ( q, f, F ) and P ( q, f + α, F ), and then apply-ing Lemma A.3 to the functions Φ and Ψ and using (2.4), we obtain the asymptotics p λ n − p µ n = (cid:18) n − ind f + ind F (cid:19) − r − α ( h ′ ) π + ℓ (cid:18) n (cid:19)! , where r := ind f − d = ( , ind f is odd,0 , ind f is even.4. Inverse problem
In this section, we will prove Theorem 1.1. The results of the previous sectionshows that if two sequences { λ n } n ≥ and { µ n } n ≥ are the eigenvalues of the prob-lems P ( q, f, F ) and P ( q, f + α, F ), then they interlace and satisfy asymptotics ofthe form p λ n = n − L + σπn + ℓ (cid:18) n (cid:19) , p λ n − p µ n = ( n − L ) − r − (cid:18) ν + ℓ (cid:18) n (cid:19)(cid:19) (4.1)where L := ind f + ind F ≥ d − − r ≥ − , ν := α ( h ′ ) π = 0 . Note also that if L = − / f = 0, and consequently r = 0. We are nowgoing to prove that these conditions are also sufficient for two sequences to be theeigenvalues of two such problems. However, unlike the case of constant boundaryconditions, in order to determine these problems uniquely, we need some additionaldata. The identity Ψ( λ ) − Φ( λ ) = αf ↓ ( λ ) χ (0 , λ ) (see Section 2) shows that thezeros of f ↓ are also zeros of Ψ − Φ. As we will see shortly, they can be chosenarbitrarily among the zeros of Ψ − Φ.Let now { λ n } n ≥ and { µ n } n ≥ be any two sequences such that they interlaceand satisfy asymptotics of the form (4.1) for some integer or half-integer L ≥ − / ν = 0 and σ , and r ∈ { , } . Define the functionsΦ( λ ) := − Y n
0) then µ n < λ n < µ n +1 (respectively, λ n < µ n < λ n +1 ) for each n ≥
0. Thus the numbers γ n defined by γ n := πνp ( λ n )Φ ′ ( λ n )Ψ( λ n )are all positive and have the asymptotics γ n = ( n − L ) d +2 r (cid:18) π ℓ (cid:18) n (cid:19)(cid:19) . By [11, Theorem 4.4], there exists a boundary value problem P ( q, f, F ) havingthe eigenvalues { λ n } n ≥ and the norming constants { γ n } n ≥ . Moreover, ind f =2 d + r ≥ F = 2 L − d − r ≥ −
1. Denote α := πν/ ( h ′ ) with h ′ defined as at the beginning of Section 2. It only remains to show that the problem P ( q, f + α, F ) has the eigenvalues µ n . But first we show that the polynomials f ↓ ( λ )and p ( λ ) coincide up to a constant factor. Arguing as in the proof of Lemma 2.1we have ∞ X n =0 λ kn p ( λ n ) γ n = ∞ X n =0 λ kn Ψ( λ n ) πνp ( λ n )Φ ′ ( λ n ) = 12 π ν i lim N →∞ Z C N λ k (Ψ( λ ) − Φ( λ )) p ( λ )Φ( λ ) d λ = 0 , where C N is the same as in that proof. Now arguing as after Lemma 2.1 weobtain that the non-leading coefficients of the polynomial ( − d p ( λ ) satisfy thesystem (2.6). Therefore f ↓ ( λ ) = h ′ p ( λ ).Denote the eigenvalues of the boundary value problem P ( q, f + α, F ) by b µ n .They coincide with the zeros of the function b Ψ( λ ) := F ↑ ( λ ) ψ ( π, λ ) − F ↓ ( λ ) ψ ′ ( π, λ ) , where ψ ( x, λ ) is defined as in (2.1). Using the results of Section 3, we obtain b Ψ( λ n ) = αf ↓ ( λ n )Φ ′ ( λ n ) γ n = πνp ( λ n )Φ ′ ( λ n ) γ n = Ψ( λ n ) , n ≥ . This and the proofs of Lemmas 2.1, A.1 and A.3 show that (cid:16) b Ψ( λ ) − Ψ( λ ) (cid:17) / Φ( λ )is an entire function satisfying the estimate b Ψ( λ ) − Ψ( λ )Φ( λ ) = O (cid:18) √ λ (cid:19) on S N C N and hence by the maximum principle on the whole plane. Then theLiouville theorem implies that this function is identically zero. Thus b Ψ( λ ) ≡ Ψ( λ )and hence b µ n = µ n , n ≥ N TWO-SPECTRA INVERSE PROBLEMS 9
Appendix A. Auxiliary results
In this appendix we prove two auxiliary lemmas used in the main body of thepaper.
Lemma A.1.
The characteristic function Φ( λ ) of P ( q, f, F ) with ind f ≥ has theinfinite product representation Φ( λ ) = − Y n
The first-order asymptoticsΦ( λ ) = λ L +1 / sin (cid:16) √ λ + L (cid:17) π + O (cid:16) λ L e | Im √ λπ | (cid:17) (A.1)was obtained in [11] (see the proof of Lemma 2.2 therein). From Hadamard’stheorem we obtainΦ( λ ) = C ∞ Y n =0 (cid:18) − λλ n (cid:19) = C Y n
2. Thenwe can combine infinite product representations for the sine and cosine functionsinto sin (cid:16) √ λ + L (cid:17) π = ( − ⌊ L ⌋ Y n = L π √ λ Y n>L − λ ( n − L ) ! . The use of the identities( − ⌊ L ⌋ = − Y n
Lemma A.2 ([22, Lemma 3.3], [21, Lemma 3.4.2]) . For functions u ( z ) and v ( z ) to admit representations of the form u ( z ) = sin πz + Aπ z z − πz + g ( z ) z , v ( z ) = cos πz − Bπ sin πzz + g ( z ) z , where g ( z ) = R π e g ( t ) cos zt d t and g ( z ) = R π e g ( t ) sin zt d t with e g , e g ∈ L [0 , π ] and R π e g ( t ) d t = 0 , it is necessary and sufficient to have the form u ( z ) = πz ∞ Y n =1 n − ( u n − z ) , u n = n − An + ℓ (cid:18) n (cid:19) ,v ( z ) = ∞ Y n =1 (cid:18) n − (cid:19) − ( v n − z ) , v n = n − − Bn + ℓ (cid:18) n (cid:19) . Now we can prove
Lemma A.3.
Let { η n } n ≥ and { ζ n } n ≥ be sequences of real numbers having theasymptotics √ η n = n − L + σπn + ℓ (cid:18) n (cid:19) , p ζ n = n − L + O (cid:18) n (cid:19) , with an integer or half-integer L ≥ − / and a real σ , and let G ( λ ) := − Y n
Using the asymptotics of η n and Lemma A.2 we obtain the representations G ( λ ) = λ L +1 / sin (cid:16) √ λ + L (cid:17) π − σλ L cos (cid:16) √ λ + L (cid:17) π + G ( λ )and G ′ ( λ ) = π λ L cos (cid:16) √ λ + L (cid:17) π + (cid:18) L + σπ + 12 (cid:19) λ L − / sin (cid:16) √ λ + L (cid:17) π + G ( λ ) , where G ( λ ) and G ( λ ) are of the form G ( λ ) = λ L Z π e G ( t ) cos (cid:16) √ λt + Lπ (cid:17) d t + O (cid:16) λ L − / e | Im √ λπ | (cid:17) and G ( λ ) = λ L − / Z π e G ( t ) sin (cid:16) √ λt + Lπ (cid:17) d t + O (cid:16) λ L − e | Im √ λπ | (cid:17) with e G ∈ L [0 , π ]. The statement of the theorem now follows from the asymptoticsof ζ n . (cid:3) N TWO-SPECTRA INVERSE PROBLEMS 11
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