Inverse spectral problems for non-local Sturm

IOP PUBLISHING

INVERSE PROBLEMS

Inverse Problems 23 (2007) 523–535

doi:10.1088/0266-5611/23/2/005

Inverse spectral problems for non-local Sturm–Liouville operators S Albeverio1,2,3,4,5, R O Hryniv6,7 and L P Nizhnik8 1

Institut f¨ur Angewandte Mathematik, Universit¨at Bonn, Wegelerstr. 6, D-53115 Bonn, Germany SFB 611 and IZKS, Bonn, Germany 3 BiBoS, Bielefeld, Germany 4 CERFIM, Locarno, Switzerland 5 Accademia di Architettura, Mendrisio, Switzerland 6 Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv, Ukraine 7 Lviv National University, 1 Universytets’ka st., 79602 Lviv, Ukraine 8 Institute of Mathematics, 3 Tereshchenkivs’ka st., 01601 Kyiv, Ukraine 2

E-mail: [email protected], [email protected] and [email protected]

Received 28 November 2006, in final form 12 January 2007 Published 5 February 2007 Online at stacks.iop.org/IP/23/523 Abstract We solve the inverse spectral problem for a class of Sturm–Liouville operators with singular non-local potentials and non-local boundary conditions. We study to what extent the operator from that class is determined by its spectrum, and point out subclasses in which the reconstruction problem from one spectrum has a unique solution.

1. Introduction The main aim of the paper is to study the inverse spectral problem for operators given by the non-local Sturm–Liouville eigenvalue problems of the form
(y) := −y

(x) + v(x)y(1) = λy(x)

(1.1)

subject to the boundary conditions y(0) = y
(1) + (y, v)L2 = 0.

(1.2)

Here v ∈ L2 (0, 1) is the non-local ‘potential’ and λ ∈ C is the spectral parameter. Problems of this form appear, e.g., in the theory of diffusion processes, where the generators of the Feller processes, or processes with Wentzell boundary conditions, usually involve non-local interactions both in the equation and the boundary conditions (see [12, 27, 28] and the references therein). Also, point interactions in three dimensions are described by nonlocal boundary conditions, see the original paper by Berezin and Faddeev [5] and a more abstract treatment in [2], as well as the references therein. For some other applications and 0266-5611/07/020523+13$30.00 © 2007 IOP Publishing Ltd Printed in the UK

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a detailed survey of the development of the theory up to 1975 see the papers of Coddington [10] and Krall [19]. The inverse spectral theory for Sturm–Liouville and Schr¨odinger operators, which aims at a determination of these operators from their related spectral data, started in the mid last century from the pioneering works by Borg [6], Levinson [22], Marchenko [23], Gel’fand and Levitan [14], Krein [21], and is now well developed for regular (i.e., locally summable) potentials. Since then these classical results have been extended and generalized in various ways in a large number of papers. In particular, the case where potentials are signed measures was treated in [4, 7, 30], Sturm–Liouville operators in impedance form with non-smooth impedance functions were attacked in [1, 3, 11, 15, 16, 25], 1/x γ -like potentials were considered in [8, 9, 13, 29, 31], and non-local boundary conditions in [20]. The inverse spectral problem for singular Sturm–Liouville operators with distributional potentials has recently been solved in [17, 18, 26]. The operator T = T (v) given by the differential expression (1.1) and the boundary conditions (1.2) can also be regarded as a singular perturbation of the operator T (0) by the ‘potential’ (·, δ1 )v + (·, v)δ1 , where δ1 is the Dirac delta-function centred at x = 1. It is a self-adjoint operator in the Hilbert space L2 (0, 1) and has a discrete spectrum λ1
λ2
· · · of multiplicity at most 2. Since T (v) is a rank-two perturbation of the unperturbed operator corresponding to v = 0, in general one would not expect that the potential v can be uniquely determined by the spectrum σ (v) := {λn }n∈N of T (v). However, we prove that for a real-valued potential v the spectrum σ (v) determines uniquely a (nonlinear) projection v of v onto some closed convex subset V of the real part L2,R (0, 1) of L2(0,1) . This enables a complete description of the set of all isospectral potentials and implies that v ∈ V is uniquely determined by σ (v), which is, e.g., the case if v has √ the L2 -norm not greater than 2π . Also, v can be uniquely reconstructed, e.g., within the sets of that coincide with algebraic polynomials or trigonometric polynomials in 
functions sin π n − 12 x and cos π nx on a subinterval (α, β) of (0, 1). The paper is organized as follows. In the next section, we study in detail the spectral properties of the operator T. Isospectral sets are characterized in section 3, and in section 4 the inverse spectral problem of reconstructing real-valued non-local potentials is solved. Finally, two reconstruction algorithms are presented in section 5. 2. Direct spectral analysis We define the operator T through the formula T y =
(y) on the domain    dom T = y ∈ W22 (0, 1)  y(0) = y
(1) + (y, v)L2 = 0 . Standard integration by parts shows that the quadratic form (T y, y)L2 is real for every y ∈ dom T , so that the operator T is symmetric. Along with T we introduce the operators T0 and T1 that are given by the differential expression −d2 /dx 2 subject to the boundary conditions y(0) = y(1) = 0 and y(0) = y
(1) = 0 respectively. The operators T0 and T1 are self-adjoint and have simple discrete 2  
spectra, namely σ (T0 ) = {π 2 n2 }n∈N and σ (T1 ) = π 2 n − 12 n∈N ; in both cases the eigenfunction corresponding to an eigenvalue z2 is sin zx. The resolvents of T0 and T1 are integral operators,  1 Gj (x, s; z)f (s) ds, j = 0, 1, (Tj − z2 )−1 f (x) = 0

Non-local Sturm–Liouville operators

whose kernels are the corresponding Green functions  1 sin zx sin z(1 − s) G0 (x, s; z) = z sin z sin z(1 − x) sin zs  1 sin zx cos z(1 − s) G1 (x, s; z) = cos z(1 − x) sin zs z cos z

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for for for for

s > x, x > s, s > x, x > s.

Lemma 2.1. The operator T is self-adjoint and has a discrete spectrum {λn }n∈N , where λ1
λ2
· · · and each eigenvalue is repeated according to its multiplicity. Moreover, T is a rank-one perturbation of the operator T0 and the spectra of the operators T and T0 weakly interlace, i.e., λn
π 2 n2
λn+1 for every n ∈ N. Proof. The operators T and T0 coincide on the common domain D := dom T ∩ dom T0    = y ∈ W22 (0, 1)  y(0) = y(1) = y
(1) + (y, v) = 0 ,

operator. Since A is the restriction of the selfand their restriction A onto
D is a symmetric  adjoint operator T0 and dim dom T0 /D = 1, it follows that A has deficiency indices (1, 1). Therefore T, being a proper symmetric extension of A, is self-adjoint, and it is a rank-one perturbation of T0 . The Krein resolvent formula shows that T is bounded below and has a discrete spectrum along with T0 . The interlacing property of the spectra follows from the variational principle. Indeed, the quadratic form t of T equals   t[f ] = (f
, f
)L2 + 2 Re f (1)(v, f )L2 on the domain

   dom t = f ∈ W21 (0, 1)f (0) = 0 ,

and the restriction of t onto functions satisfying the condition f (1) = 0 coincides with the quadratic form t0 of the operator T0 .  The resolvent of the operator T for the points λ = z2 in the resolvent set of both T and T0 can be written in the following way. Put  sin zx sin z 1 ϕ(x; z) := G0 (x, s; z)v(s) ds; − z z 0 then ϕ(·; z) is a unique (up to a scalar multiple) solution of the equation
(y) = z2 y satisfying the initial condition y(0) = 0. Denote by b(f ) := f
(1) + (f, v)L2 the boundary functional determining the operator T; then z2 is an eigenvalue of T in R\σ (T0 ) and ϕ(·; z) is a corresponding eigenfunction if and only if b(ϕ(·; z)) = 0, i.e., if z is a zero of the function  1 sin zs d(z) := b(ϕ(·, ; z)) = cos z + (v(s) + v(s)) ds z 0   sin z 1 1 − G0 (x, s; z)v(s)v(x) ds dx. z 0 0 Take an arbitrary g ∈ L2 (0, 1); then a generic solution of the equation −y

(x) + y(1)v(x) = z2 y + g satisfying the initial condition y(0) = 0 has the form  1 G0 (x, s; z)g(s) ds + cϕ(x; z) y(x) = 0

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with an arbitrary c ∈ C. The condition b(y) = 0 determines this c uniquely, and we finally get z ϕ(x; z)(g, ϕ(·; z)). (T − z2 )−1 g(x) = (T0 − z2 )−1 g(x) + sin zd(z) The function d is the characteristic function of the operator T, cf theorem 2.2 describing some spectral properties of the operator T. Theorem 2.2. (i) Every eigenvalue of the operator T is a squared zero of its characteristic function d and, conversely, every squared zero of d is an eigenvalue of T. The number π 2 n2 , n ∈ N, is an eigenvalue of T if and only if  1 vˆ n := 2 v(s) sin π ns ds = (−1)n+1 2π n, (2.1) 0

and this relation is equivalent to d(π n) = 0. (ii) All eigenvalues z2 not in the spectrum of T0 are simple, and simple are the corresponding zeros z of d (except for the case where z = 0, which is then a zero of even order of d). If π 2 n2 for some n ∈ N is an eigenvalue of T, then this eigenvalue is multiple if and only if  1 1 G1 (x, s; π n)v(s)v(x) ds dx = 0; (2.2) 0

0

in this and only in this case the number π n is a multiple zero of d. (iii) The multiplicity of a non-zero eigenvalue z2 of the operator T equals the order of the corresponding zero z of the characteristic function d, and both do not exceed 2. If z = 0 is an eigenvalue of T, then the order of z = 0 as a zero of d is 2. Before proceeding with the proof of the theorem, we derive equivalent forms of some of the above  relations. Namely, if vˆ k denotes the kth sine Fourier coefficient of the function ˆ k sin π kx, then straightforward calculations give v(x) := ∞ k=1 v ∞

ϕ(x; z) =

sin zx sin z vˆ k sin π kx + , z z k=1 z2 − π 2 k 2

ak sin z , d(z) = cos z + 2 2z k=1 z − π 2 k 2

(2.3)

ak := |ˆvk |2 + (−1)k 2π k(ˆvk + vˆ k ) = |ˆvk + (−1)k 2π k|2 − (2π k)2 .

(2.4)

∞

where Taking the limit as z → π n in (2.3), we find that d(π n) = (−1)n |ˆvn + (−1)n 2π n|2 /(2π n)2 ,

(2.5)

which shows that d(π n) = 0 if and only if (2.1) holds. Proof of theorem 2.2. (i) We have shown above that, for z that is not equal to ±π n, n ∈ N, d(z) vanishes if and only if z2 is an eigenvalue of the operator T. Assume now that λ = π 2 n2 for some n ∈ N is an eigenvalue of T. If there is an eigenfunction y such that y(1) = 0, then y is also an eigenfunction of T0 corresponding to the same eigenvalue and thus y(x) = C sin π nx. The boundary condition y
(1) + (y, v) = 0 then yields vˆ n = (−1)n+1 2π n, i.e., (2.1). If y is an eigenfunction of T that does not vanish at x = 1, then we can normalize y by the condition y(1) = 1. In this case the function y equals y(x) = a sin π nx + u(x; π n)

(2.6)

Non-local Sturm–Liouville operators

for some constant a, where

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1

u(x; π n) := −

G1 (x, s; π n)v(s) ds

(2.7)

0

is a particular solution of the equation −y

= π 2 n2 y − v satisfying the initial condition y(0) = 0. Equating y of (2.6) to 1 at x = 1 gives the relation  1 (−1)n+1 vˆ n 1 sin π nsv(s) ds = 1 = u(1; π n) = − π n cos π n 0 2π n i.e., relation (2.1). We have thus shown that if π 2 n2 is an eigenvalue of T, then relation (2.1) holds. Conversely, a direct verification shows that (2.1) implies that sin π nx is an eigenfunction of T corresponding to the eigenvalue π 2 n2 . Therefore π 2 n2 is an eigenvalue of T if and only if (2.1) holds, i.e., if and only if d(π n) = 0. This completes the proof of (i). (ii) Assume that λ = z2 is an eigenvalue of T that is not in the spectrum of T0 , i.e., that λ is not equal to π 2 n2 for n ∈ N. Then this eigenvalue is simple as otherwise there would exist an eigenfunction y with y(1) = 0 and then y would also be an eigenfunction of T0 corresponding to λ, contrary to the assumption. We also note that such a z, once ˙ not equal to zero, is a simple zero of d. Indeed, otherwise d(z) = 0, which implies that the function ψ(·; z) := ϕ(·; ˙ z) satisfies the boundary conditions ψ(0) = b(ψ) = 0 and the relation (T − z2 )ψ(·; z) = 2zϕ(·; z). Then ϕ(x; z) and ψ(·; z) form a chain of eigen- and associated functions of the operator T corresponding to the eigenvalue z2 , which is impossible for self-adjoint operators. Since d is an even function, d(0) = 0 implies that z = 0 is a zero of d of even order. Assume now that λ = π 2 n2 is an eigenvalue of T of multiplicity at least 2. Then there is an eigenfunction satisfying the terminal condition y(1) = 1, and, adding a suitable multiple of the eigenfunction sin π nx if necessary, we get y
(1) = 0. Therefore such an eigenfunction must be equal to the function u(·; π n) of (2.7). By construction, the function u(·; π n) satisfies the equation −y

= z2 y − v and the boundary conditions y(0) = y
(1) = 0; moreover,  1 (−1)n+1 vˆ n 1 =1 sin π nsv(s) ds = u(1; π n) = − π n cos π n 0 2π n by (2.1). Thus u(·; π n) is an eigenfunction of T only if it satisfies the boundary condition b(y) := y
(1) + (y, v) = 0, i.e., if (u(·; π n), v)L2 = 0, and this is precisely relation (2.2). Conversely, if λ = π 2 n2 is an eigenvalue of T and (2.2) holds, then u(·; π n) is an eigenfunction of T that is linearly independent of the eigenfunction sin π nx; therefore λ = π 2 n2 is a multiple eigenvalue of T . Assume that λ = π 2 n2 is an eigenvalue of T. Using formula (2.3), we get   n 3a a (−1) k n  ˙ = ˙ n) := lim d(z)  − d(π , (2.8) z→πn 2π n k=n π 2 (n2 − k 2 ) (2π n)2 where ak are given by (2.4); in particular, an = −(2π n)2 on account of (2.1). On the other hand, straightforward but tedious calculations of the numbers  1 1 G1 (x, s; π n) sin π ks sin π lx ds dx ckl := 0

0

on account of the equality vˆ n = (−1)n+1 2π n result in    1 1 ∞ ak 1  − 3 G1 (x, s; π n)v(s)v(x) ds dx = vˆ k vˆ l ckl = 2 (k 2 − n2 ) 2 π 0 0 k=n k,l=1 ˙ n). = (−1)n+1 π nd(π

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˙ n) = 0 if and only if (2.2) holds, i.e., if and only if λ = π 2 n2 is a multiple Therefore d(π eigenvalue of the operator T. (iii) It follows from the interlacing property of the spectra of the operators T and T0 that no eigenvalue of T has multiplicity greater than 2. Items (i) and (ii) now imply that the order of every zero z of d is not less than the multiplicity of z2 as an eigenvalue of the operator T. We show next that in fact this order of z coincides with the multiplicity of the eigenvalue z2 provided z = 0. Indeed, denote by {zk } the set of all zeros of d in the domain C+ := {z ∈ C\{0} | −π/2 < arg z
π/2} ∪ {0}

(2.9)

repeated according to their orders (if z = 0 is a zero of d of order 2m, then it is repeated m times among the zeros in C+ ). The standard arguments based  Rouch´e theorem show
on the that the zeros zk can be enumerated in such a way that zk = π k − 12 + o(1) as k → ∞. Since the operators T and T0 can have only finitely many common eigenvalues, by lemma 2.1 the eigenvalues λ1
λ2
· · · of T satisfy π 2 (k − 1)2 < λk < π 2 k 2 for all sufficiently large k. It follows that λk = zk2 for all k large enough. Thus there exists a one-to one correspondence  between the eigenvalues λk of T and the zeros zk of d in C+ , and (iii) follows. Example 2.3. Consider a non-local potential of the form v(x) = 2π sin π x + π sin 2π x. Then vˆ 1 = 2π, vˆ 2 = π , and vˆ n = 0 for all n  3. Thus relations (2.1) and (2.2) hold with n = 1, and the number λ = π 2 is a multiple eigenvalue of the corresponding operator T. Two linearly independent eigenfunctions are y1 (x) = sin π x and y2 (x) = x cos π x + sin 2π x/(3π ). Theorem 2.4. The eigenvalues λ1
λ2
· · · satisfy the asymptotic distribution 
 λn = π n − 12 + µn /n

(2.10)

for some sequence (µn )n∈N in
2 (N).

√ Proof. By theorem 2.2 the numbers zn := λn , n ∈ N, give all the zeros of the characteristic function d in the domain C+ of (2.9) and are repeated according to their orders (recall the above special definition of the order of z = 0 in C+ if z = 0 is a zero). Since d is an even function, it suffices to consider only its zeros in C+ . We start with observing that the zeros z˜ n of the function  1 ˜ := cos z + 2 sin zx Re v(x) dx d(z) z 0 in C+ have the required representation (see [24, lemma 3.4.2]). We denote by γn the circle of radius rn about the point z˜ n , with rn ∈ (n−2 , 2n−2 ) chosen so that γn does not intersect the set π Z. Then n n 1
sup max sup max < ∞, sup max 2 2 2 z∈γ z∈γ z∈γ n |π k − z | n |π k + z| n,k∈N n |π k − z| n,k∈N n,k∈N and the spectral theorem combined with the dominated convergence theorem implies that    2n|ˆv |2  k   2 −1 lim max n|((T0 − z ) v, v)| = lim max   = 0. n→∞ z∈γn n→∞ z∈γn  π 2 k 2 − z2  k∈N

Therefore,

   sin z  2 −1  max  ((T0 − z ) v, v)L2  = o(n−2 ), z∈γn z

Non-local Sturm–Liouville operators

529

while, on the other hand, for all sufficiently large n,   ˜  1  dd(z)   rn .  ˜ min |d(z)|  min rn  z∈γn z:|z−˜zn |
rn dz  2 Applying the Rouch´e theorem, we see that there is n0 such that, for all n  n0 , the with
zn bythe disc surrounded by γn contains exactly one zero of d, which coincides  zn − π n − 1 
derived in the proof of theorem 2.2(iii). As rough asymptotics of z k 2  
z˜ n − π n − 1  + rn , we see that the zeros zn have the required representation for all n with 2  |n|  n0 . 3. Isospectral non-local potentials The characteristic function d = d(v) of the operator T (v) is an even entire function of exponential type 1 satisfying the relation d(iξ ) lim =1 ξ →∞ cos iξ and therefore can uniquely be given by the Hadamard canonical product in its zeros. Namely, the characteristic function d equals  λk − z 2 (3.1) d(z) =
 , 1 2 2 k∈N π k − 2 where {λk } is the set of all eigenvalues of the operator T counting multiplicity. We also note the following property of d. Lemma 3.1. The characteristic function d of the operator T (v), v ∈ L2 (0, 1), is uniquely determined by its values at the points π k, k ∈ N. Proof. In view of (2.5), the values of d at the points π k, k ∈ N, uniquely determine |ˆvk + (−1)k 2π k| and thus the coefficients ak of (2.4) in the representation (2.3) of d.  The main object of this section is isospectral non-local potentials, i.e., those v (1) and v (2) in L2 (0, 1), for which the spectra of the operators T (v (1) ) and T (v (2) ) coincide. Assume that v (1) and v (2) are isospectral. Since by theorem 2.2 the eigenvalues of T (v (1) ) and T (v (2) ) are squared zeros of the corresponding characteristic functions d1 and d2 , and the latter are uniquely determined by their zeros via (3.1), we conclude that d1 ≡ d2 . Equating the values of d1 and d2 at the points π n, n ∈ N, we derive from (2.5) that  (1)    vˆ + (−1)n 2π n = vˆ (2) + (−1)n 2π n. (3.2) n

n

Conversely, if some v (1) and v (2) in L2 (0, 1) are such that (3.2) holds for all n ∈ N, then by (2.5) the values of the characteristic functions d(v (1) ) and d(v (2) ) of the operators T (v (1) ) and T (v (2) ) at the points π n, n ∈ N, coincide. Thus d(v (1) ) ≡ d(v (2) ) by lemma 3.1 and v (1) and v (2) are isospectral. For potentials that take only real values (respectively only purely imaginary values) this can be specified as follows. Proposition 3.2. (i) Assume that the potentials v (1) and v (2) are real-valued. Then they are isospectral if and only if, for every n ∈ N, either vˆ n(1) = vˆ n(2) or vˆ n(1) + vˆ n(2) = (−1)n+1 4π n. (ii) Assume that the potentials v (1) and v (2) take only purely imaginary values. Then they are isospectral if and only if, for every n ∈ N, either vˆ n(1) = vˆ n(2) or vˆ n(1) = −ˆvn(2) .

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Example 3.3. The set of real-valued non-local potentials isospectral to v ≡ 0 can be parametrized by the family F of finite subsets of N, with the potential vS corresponding to S ∈ F given by vS (x) := 4π (−1)n+1 n sin π nx. n∈S

Proposition 3.2 (i) implies that the Fourier coefficients of every two real-valued non-local isospectral potentials coincide starting from some index n ∈ N and that, for every r > 0, the ball Br := {u ∈ L2 (0, 1) | u
r} contains only finitely many real-valued potentials which are isospectral to a given one. 4. The inverse spectral analysis We denote by L2,R (0, 1) the real Hilbert space of real-valued functions in L2 (0, 1). We shall further extend the inverse spectral analysis for non-local potentials in L2,R (0, 1) started in the previous section. The crucial observation is that relation (2.5) shows that the kth Fourier coefficient vˆ k of a real-valued v is a solution of the quadratic equation (ˆvk + (−1)k 2π k)2 = (−1)k (2π k)2 d(π k), i.e., vk equals one of the two numbers vˆ k± := (−1)k+1 2π k[1 ±



(−1)k d(π k)].

(4.1)

(4.2)

Clearly, vˆ k = vˆ k− for all k large enough. This observation motivates the following definition. of L2,R (0, 1) consisting of those functions v(x) =  We denote by V the subset ˆ n sin π nx, for which (−1)n+1 vˆ n
2π n for all n ∈ N. It is clear that V is a closed n∈N v convex set in L2,R (0, 1) not containing √ any line {tv | t ∈ R}. Moreover, all functions that

v


2π belong to V , as easily follows from the Parseval v ∈ L2,R (0, 1) such L 2  equality v 2 = 12 n∈N |ˆvn |2 . More generally, if the sine Fourier series of a v ∈ L2,R (0, 1) √  starts with vˆ N , i.e., if v(x) = nN vˆ n sin π nx, and v L2
2N π , then v ∈ V .  Definition 4.1. Let v be a function in L2,R (0, 1) with the sine Fourier  series n∈N vˆ n sin π nx. The projection of v on V is the function v˜ with the sine Fourier series n∈N v˜ n sin π nx, where  if (−1)n+1 vˆ n
2π n, vˆ n v˜ n = otherwise. (−1)n+1 4π n − vˆ n The induced mapping  : v → v˜ =: v is a nonlinear projector of L2,R (0, 1) on V , i.e., v = v if v ∈ V , so that 2 = . Since vˆ k → 0 as k → ∞ for every v ∈ L2,R (0, 1), for such v the function v − v is a trigonometric polynomial in {sin π nx}. Also, |˜vk |
|ˆvk |, and hence v
v ; moreover, v = v if and only if v ∈ V . √ The above considerations imply that the set V contains the ball of radius 2π , but otherwise this set is difficult to describe. However, for many subspaces S the projection v ∈ V of v ∈ S uniquely determines this v. An important case when this occurs is given by the following statement. Lemma 4.2. Assume that S is a linear subset of L2,R (0, 1) not containing any non-zero trigonometric polynomial in {sin π nx}n∈N . Then the projector  is injective on S .

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Proof. If v1 and v2 in S are such that v1 = v2 , then the function v1 − v2 = (v1 − v1 ) − (v2 − v2 ) must be a trigonometric polynomial in {sin π nx}n∈N . Thus  v1 = v2 . Example 4.3. Take an arbitrary interval (α, β) ⊂ (0, 1) and consider the set S of all functions from L2,R (0, 1), whose restrictions onto (α, β) are (i) algebraic polynomials; (ii) trigonometric polynomials in cos  sx,
s ∈ R;  (iii) trigonometric polynomials in sin π n − 12 x . In each of these cases, S verifies the assumptions of lemma 4.2. The importance of the projector  in the spectral analysis of the operators T (v) is clarified by the following theorem. Theorem 4.4. Assume that v1 and v2 belong to L2,R (0, 1). Then v1 and v2 are isospectral if and only if v1 = v2 . Proof. The spectrum σ (v) of T (v) determines uniquely the characteristic function d of T (v) via (3.1) and thus the numbers vˆ k± of (4.2). Therefore σ (v1 ) = σ (v2 ) implies that the sine Fourier series of v1 and v2 coincide, i.e., that v1 = v2 . Conversely, if v1 = v2 , then the numbers vˆ k± in (4.2) constructed for the two functions coincide, and thus the characteristic functions d(v1 ) and d(v2 ) of the operators T (v1 ) and T (v2 ) take the same values at the points π k. Then d(v1 ) ≡ d(v2 ) by lemma 3.1, so that v1 and v2 are isospectral potentials.  Theorem 4.4 and properties of  yield several immediate corollaries. Corollary 4.5. Assume that v belongs to V . Then v is uniquely reconstructed from the spectrum of the operator T (v). Corollary 4.6. Assume that a set S verifies the assumptions of lemma 4.2. Then a non-local potential v ∈ S is uniquely determined by the spectrum σ (v) of the corresponding operator T (v). Corollary 4.7. Among all potentials in L2,R (0, 1) that are isospectral to a given v ∈ L2,R (0, 1), the smallest norm has v. Finally we complete the characterization of the possible spectra of operators T (v) when v runs through the set V . Theorem 4.8. Assume that a sequence λ1
λ2
· · · is such that it weakly interlaces the sequence (π 2 n2 )n∈N , i.e., λn
π 2 n2
λn+1 for all n ∈ N, and satisfies relation (2.10) with some (µn ) ∈
2 . Then there exists a unique v ∈ V such that the spectrum of the operator T (v) coincides with {λn }n∈N counting multiplicities. Proof. Given the sequence (λn )n∈N as in the theorem, we first construct a function  d via (3.1), then for every n ∈ N put vˆ n := vˆ n− with vˆ n− of (4.2), and finally take v := n∈N vˆ n sin π nx as the non-local potential looked for. We first claim that the numbers vˆ n so constructed are real, satisfy the inequality (−1)n+1 vˆ n
2π n for all n ∈ N, and form an
2 -sequence, so that v∈V. The interlacing property of λn and the explicit formula for the function d show that (−1)n d(π n)  0 for all n ∈ N. Therefore the numbers vˆ n± of (4.2) are real and

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(−1)n+1 vˆ n−
2π n. Next, the asymptotic form of λn and [24, lemma 3.4.2] imply that the canonical product d of (3.1) admits the representation  1 sin zt dt f (t) d(z) = cos z + z 0 for some f ∈ L2 (0, 1). It follows that d(π n) = (−1)n + fˆ (π n)/(π n), where fˆ (π n) is the nth sine Fourier coefficient of f . Substituting this expression into formula (4.2), we see that the numbers vˆ n− form an
2 -sequence and thus are sine Fourier coefficients of some function v in V . It remains to prove that the spectrum of the operator T (v) with the v found coincides with the sequence λn we have started with. To this end we note that the characteristic function d1 of the operator T (v) coincides with d at the points π n, n ∈ N, by construction. By lemma 3.1 d ≡ d1 , and thus zeros of d and d1 coincide. It follows that the eigenvalues of T (v) are  λn , n ∈ N. 5. The reconstruction procedure The analysis of the previous section shows that the following inverse spectral problem is well posed. Inverse problem 1. Given the spectrum σ (v) of an operator T (v) for some v ∈ V , find the non-local potential v. Algorithm of solution of the inverse problem 1. Step 1. Given σ (v), construct the function d via (3.1). Step 2. Calculate the values d(π n), n ∈ N. Step 3. For every n ∈ N, solve the quadratic equations of (4.1) for vˆ n , taking the solution that satisfies the relation (−1)n+1 vˆ n
2π n.  Step 4. Put v = n∈N vˆ n sin π nx.
2 Example 5.1. Let λ1 = π 2 and λn = π 2 n − 12 for all n  2. Then d(z) =

z2 − π 2 cos z, z2 − π 2 /4

so that d(π k) = (−1)k (k 2 − 1)/(k 2 − 1/4), vˆ 1 = 2π by (2.1), and    2−1 k  vˆ k = (−1)k+1 2π k 1 − k 2 − 1/4 for k  2. Next we discuss a truncated inverse problem, where only finitely many eigenvalues of T (v) are given, and v is to be found within a special class. Inverse problem 2. Assume that given are N real numbers λ1
λ2 · · ·
λN such that λk
π 2 k 2
λk+1 for k = 1, 2, . . . , N. Find a non-local potential vN of the form vN (x) =

N k=1

vˆ k sin π kx

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such that the first N eigenvalues of the operator T (vN ) coincide with the given numbers λn , n = 1, 2, . . . , N. If vN with the required properties exists, then the√characteristic function √ √ dN of the operator T (vN ) must satisfy the equalities dN ( λ1 ) = dN ( λ2 ) = · · · = dN ( λN ) = 0. Recalling the representation (2.3) of dN , we can regard these equalities as a linear system of N equations for a1 , a2 , . . . , aN of (2.4) of the form N k=1

√  ak = 2 λj cot λj , − λj

j = 1, 2, . . . , N

π 2k2

(5.1)

(observe that vˆ n = an = 0 for n > N). If one of λj is equal to π 2 n2 for some n ∈ {1, 2, . . . , N } (necessarily j = n − 1 or j = n), then (2.5) immediately produces vˆ n = (−1)n+1 2π n and an = −(2π n)2 . If this number π 2 n2 appears twice among λj (i.e., if λn = λn+1 = π 2 n2 ), then we must augment the above system with the equation d˙N (π n) = 0, which by (2.8) is equivalent to the relation ak = 3. (5.2) 2 2 π (k − n2 ) k=n The sine Fourier coefficients vˆ 1 , vˆ 2 , . . . , vˆ N of dN are then found from a1 , a2 , . . . , aN by solving the corresponding quadratic equations. Algorithm of solution of the inverse problem 5. Step 1. For every n ∈ N such that π 2 n2 ∈ an = −(2π n)2 .



 λ21 , . . . , λ2N , put vˆ n = (−1)n 2π n and

Step 2. For every n ∈ N such that λn = π 2 n2 = λn+1 , write equation (5.2). Step 3. For every λj not of the form π 2 n2 write equation (5.1). Step 4. Solve the so-formed linear system of N equations for a1 , . . . , aN . Step 5. For k = 1, . . . , N, determine vˆ k from equation (2.4) and put v =

N k=1

vˆ k sin π kx.

Example 5.2. We fix λ1 < 0 and want it to be an eigenvalue of T (v) with v(x) = vˆ N sin π N x. Since a1 = · · · = aN−1 = 0, the linear system reduces to the single equation for aN , namely  √ aN = 2 λ cot λ1 , 1 π 2 N 2 − λ1  √ √ so that aN = 2(π 2 N 2 + |λ1 |) |λ1 |/tanh |λ1 |, and vˆ N = (−1)N+1 2π N ± aN + (2π N )2 . Example 5.3. Let N = 4 and λ1 = λ2 = π 2 , λ3 = λ4 = (3π )2 . Then theorem 2.2 and (2.1) ˙ )=0 yield vˆ 1 = 2π and vˆ 3 = 6π , so that a1 = −(2π )2 and a3 = −(6π )2 . The equations d(π ˙ and d(3π ) = 0 produce the system ak ak = 3π 2 , = 3π 2 , 2 2−9 k − 1 k k=1 k=3 solving which we find that a2 = 475π 2 /32 and a4 = 1225π 2 /9, and then calculate vˆ 2 and vˆ 4 . Acknowledgments The authors express their gratitude to DFG for financial support of the projects 436 UKR 113/79 and 436 UKR 113/84. The second author gratefully acknowledges the financial

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support of the Alexander von Humboldt Foundation. The second and the third authors thank the Institute for Applied Mathematics of Bonn University for the warm hospitality.

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