The “action” Variable Is Not an Invariant for the Uniqueness in the Inverse Scattering Problem

We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking S-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szegö class; on the other hand the transmission coefficient remains the same. This implies the statement given in the title. 1. Jacobi matrices of Szegö class. Direct scattering — Bernstein–Szegö type theorem As it is well known in the theory of completely integrable systems the absolute values of a reflection coefficient have played the role of the “action” variables and the arguments of this function have meaning of the “angle” variables (in this case we think on the Toda lattice as on an integrable system) [4]. Combining results of our previous works [2] (see also [3]) and [5], we give a wide set of examples where two reflection coefficients, having the same absolute values, possess completely different properties: the first one uniquely determines a Jacobi matrix of Szeqö class and the second one does not. Note that the proof of the main theorem in [2] (and, therefore, our result) essentially uses the analysis of [1] on regularization of so called Arov– singular matrix functions. Let J be a Jacobi matrix defining a bounded self–adjoint operator on l(Z): Jen = pnen−1 + qnen + pn+1en+1, n ∈ Z, (1) where {en} is the standard basis in l(Z), pn > 0. The resolvent matrix–function is defined by the relation R(z) = R(z, J) = E∗(J − z)−1E , (2) where E : C → l(Z) is such that E [ c−1 c0 ] = e−1c−1 + e0c0. This matrix–function possesses an integral representation R(z) = ∫ dσ x− z (3) with a 2×2 matrix–measure having a compact support on R. J is unitary equivalent to the multiplication operator by an independent variable on Ldσ = { f = [ f−1(x) f0(x) ] : ∫ f∗ dσ f < ∞ } . Date: January 18, 2002. 1 2 A. KHEIFETS AND P. YUDITSKII The spectrum of J is called absolutely continuous if the measure dσ is absolutely continuous with respect to the Lebesgue measure on the real axis, dσ(x) = ρ(x) dx. (4) It is natural to ask how properties of coefficients of J are reflected on its spectral properties. One is especially interested in J ’s “close” to the “free” matrix J0 with constant coefficients, pn = 1, qn = 0 (so called Chebyshev matrix). Let us mention that J0 has the following functional representation, besides the general one mentioned above. The resolvent set of J0 is the domain C̄ \ [−2, 2]. Let z(ζ) : D → C̄ \ [−2, 2] be a uniformization of this domain, z(ζ) = 1/ζ + ζ. With respect to the standard basis {t}n∈Z in L = {f(t) : ∫ T |f | dm}, the matrix of the operator of multiplication by z(t), t ∈ T, is the Jacobi matrix J0, since z(t)t = tn−1 + t. We say that J with absolutely continuous spectrum [−2, 2] is of Szegö class if its spectral density (4) satisfies log det ρ(z(t)) ∈ L. (5) Theorem 1.1. Let J be of Szegö class. Then pn → 1, qn → 0, n → ±∞. (6) Moreover, there exist generalized eigenvectors pne (n− 1, t) + qne(n, t) + pn+1e(n+ 1, t) = z(t)e(n, t) pne −(−n, t) + qne(−n− 1, t) + pn+1e(−n− 2, t) = z(t)e−(−n− 1, t) (7) such that the following asymptotics hold true s(t)e±(n, t) =s(t)t + o(1), n → +∞ s(t)e±(n, t) =t + s∓(t)t −n−1 + o(1), n → −∞ (8) in L. Thus the eigenvectors of J behave asymptotically as the eigenvectors of J0 (later we make more precise statement). The matrix formed by the coefficients of (8) S(t) = [ s− s s s+ ] (t), t ∈ T, (9) is called the scattering matrix of J . It is unitary–valued, possesses the symmetry property S∗(t̄) = S(t) and the following analytic property: s(t) is an outer function. In what follows every matrix–function of the form (9) with the above listed properties is called a scattering matrix. Of course we have a good reason for this, since with every matrix S(t) of this kind one can associate a Jacobi matrix J whose scattering matrix (associated to J according to Theorem 1.1) is the initial matrix– function S(t). However, S(t), generally speaking, does not determine J uniquely. THE “ACTION” VARIABLE IS NOT AN INVARIANT FOR UNIQUENESS 3 To clarify all above statements we need some notation and definitions. First of all for a given functions s± we define the metric ||f ||2s± = 1 2 〈[ 1 s±(t) s±(t) 1 ] [ f(t) t̄f(t̄) ] , [ f(t) t̄f(t̄) ]〉 =〈f(t) + t̄(s±f)(t̄), f(t)〉, f ∈ L, and we denote by L2dm,s± or L 2 s± (for shortness) the closure of L 2 with respect to this new metric. The following relations set a unitary map from L2s+ to L 2 s− : [ sf sf− ] (t) = [ s 0 s+ 1 ] (t) [ f(t) t̄f(t̄) ] = [ 1 s− 0 s ] (t) [ t̄f−(t̄) f−(t) ] . (10) Moreover, in this case, ||f+||2s+ = ||f||s− = 1 2 {||sf|| + ||sf−||2}. It is worth to give a scalar variant of relations between f ∈ L2s+ and f− ∈ L2s− : s(t)f∓(t) = t̄f±(t̄) + s±(t)f ±(t). Theorem 1.2. J is a Jacobi matrix of Szegö class with the spectrum E = [−2, 2] if and only if J possesses the scattering representation, i.e.: there exists a unique matrix–function S(t) of the form (9) (with the listed properties) and a unique pair of Fourier transforms F± : l(Z) → L2s± , (F±Jf)(t) = z(t)(F±f)(t), (11) determining each other by the relations s(t)(F±f)(t) = t̄(F∓f)(t̄) + s∓(t)(Ff)(t), (12) and having the following analytic properties sF(l(Z±)) ⊂ H, (13) and asymptotic properties e±(n, t) = t + o(1) in L2s± , n → +∞, (14) where e(n, t) = (Fen)(t), e−(n, t) = (Fe−n−1)(t), with {en} being the standard basis in l(Z). We point out that asymptotic relations (8) and (14) are equivalent, moreover (14) directly shows that the eigenvectors of J0 asymptotically are the eigenvectors of J . 4 A. KHEIFETS AND P. YUDITSKII 2. Uniqueness and Completeness Before we proceed with the uniqueness theorem, we show how to construct at least one J with the given scattering matrix S(t). Consider the space H s+ = Ls+ H, and introduce the Hankel operator Hs+ : H → H, Hs+f = P+ t̄(s+f)(t̄), f ∈ H, where P+ is the Riesz projection from L 2 onto H. This operator determines the metric in H s+ : ||f ||2s+ =〈f(t) + t̄(s+f)(t̄), f(t)〉 =〈(I +Hs+)f, f〉, ∀f ∈ H. Theorem 2.1. Let S(t) be a scattering matrix, i.e., the matrix of the form (9) with listed properties. Then the space H s+ is a space of holomorphic functions with a reproducing kernel. Moreover, the reproducing vector ks+ : 〈f, ks+〉 = f(0), ∀f ∈ H s+ , is of the form ks+ = (I +Hs+)1 := lim ǫ→0+ (ǫ+ I +Hs+)1 in L2s+ . (15) Put Ks+(t) = ks+(t)/ √ ks+(0). Then the system of functions {tKs+t2n(t)}n∈Z forms an orthonormal basis in L2s+ . With respect to this basis, the multiplication operator by z(t) is a Jacobi matrix J = J [s+] of Szegö class. Moreover, the initial S(t) serves as the scattering matrix–function, associated with given J by Theorem 1.2, and the relations F(en) = tKs+t2n(t) determine uniquely corresponding Fourier transforms. Let us fix the notation J [s+] for the Jacobi matrix associated with S(t) by Theorem 2.1. On the other hand, the system of functions {tKs−t2n(t)}n∈Z forms an orthonormal basis in Ls− , and we can define a Jacobi matrix J̃ = J [s−] by the relation z(t)ẽ(n, t) = p̃nẽ (n− 1, t) + q̃nẽ(n, t) + p̃n+1ẽ(n+ 1, t), where {ẽ+(n, t)} is the dual system to the system {tKs−t2n(t)} (see (12)), i.e.: s(t)ẽ(−n− 1, t) := t̄Ks−t2n(t̄) + s−(t)tKs−t2n(t). None guarantees that operators J [s+] and J [s−] are the same (see beginning of the next section). However, if J [s+] = J [s−], then the uniqueness theorem takes place. Theorem 2.2. A scattering matrix S(t) determines a Jacobi matrix J of Szegö class in a unique way if and only if the following relations take place s(0)Ks±(0)Ks∓t−2(0) = 1. (16) THE “ACTION” VARIABLE IS NOT AN INVARIANT FOR UNIQUENESS 5 Of course it is hard to check identities, especially using computer simulation, but, in fact, (16) has a specific approximating meaning. Let us return to the matrix J [s−]. According to (13) the space F(l(Z+)) is a subspace of L2s+ consisting of holomorphic in D functions. In the given case we can even specify this space in the form F(l(Z+)) = Ĥ s+ := {f ∈ L 2 s+ : sf ∈ H }. (17) Every function f from H s+ possesses the property sf ∈ H, but this means only inclusion: Ĥ s+ ⊃ H 2 s+ (18) The meaning of (16) is that every function from Ĥ s+ can be approximated by functions from H in L2s+–norm (H 2 is dense in Ĥ s+). In fact, it is enough to prove that we can approximate just two functions ẽ(0, t) and ẽ(1, t) from Ĥ s+ . This would guarantee (16), completeness and the uniqueness theorem.