On a Discrete Inverse Problem for Two Spectra

Abstract

A version of the inverse spectral problem for two spectra of finite-order real Jacobi matrices (tridiagonal symmetric matrices) is investigated. The problem is to reconstruct the matrix using two sets of eigenvalues: one for the original Jacobi matrix and one for the matrix obtained by deleting the last row and last column of the Jacobi matrix.

1. Introduction

The Jacobi matrices (tridiagonal symmetric matrices) appear in variety of applications. A distinguishing feature of the Jacobi matrices from others is that they are related to certain three-term recursion equations (second-order linear difference equations). Therefore, these matrices can be viewed as the discrete analogue of Sturm-Liouville operators, and their investigation have many similarities with Sturm-Liouville theory [1].

An 𝑁×𝑁 (real) Jacobi matrix 𝐽 is a matrix of the form𝑏𝐽=⁰𝑎₀𝑎0⋯000₀𝑏₁𝑎₁⋯0000𝑎₁𝑏₂⋯000⋮⋮⋮⋱⋮⋮⋮000…𝑏_𝑁−3𝑎_𝑁−30000⋯𝑎_𝑁−3𝑏_𝑁−2𝑎_𝑁−2000⋯0𝑎_𝑁−2𝑏_𝑁−1, (1.1) where for each 𝑛,𝑎_𝑛 and 𝑏_𝑛 are arbitrary real numbers such that 𝑎_𝑛 is different from zero:𝑎_𝑛,𝑏_𝑛∈ℝ,𝑎_𝑛≠0. (1.2) Let 𝐽₁ be the truncated matrix obtained by deleting from 𝐽 the last row and last column:𝐽₁=𝑏₀𝑎₀𝑎0⋯00₀𝑏₁𝑎₁⋯000𝑎₁𝑏₂⋯00⋮⋮⋮⋱⋮⋮000…𝑏_𝑁−3𝑎_𝑁−3000⋯𝑎_𝑁−3𝑏_𝑁−2. (1.3) Denote the eigenvalues of the matrices 𝐽 and 𝐽₁ by 𝜆₁,…,𝜆_𝑁 and 𝜇₁,…,𝜇_𝑁−1, respectively. The (finite) sequences {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 are called the two spectra of the matrix 𝐽.

The subject of the present paper is the solution of the inverse problem consisting of the following parts.(i)Is the matrix 𝐽 determined uniquely by its two spectra?(ii)To indicate an algorithm for the construction of the matrix 𝐽 from its two spectra;(iii)To find necessary and sufficient conditions for two given sequences of real numbers {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 to be the two spectra for some matrix of the form (1.1) with entries from class (1.2).

This problem was solved earlier in [2, 3]. In the present paper we offer another and more effective, as it seems to us, method of solution for this problem.

Other versions of the inverse problem for two spectra are investigated in [1, 4–9].

The paper consists, besides this introductory section, of two sections. Section 2 is auxiliary and presents briefly the solution of the inverse problem for finite Jacobi matrices in terms of the eigenvalues and normalizing numbers. A solution to this problem is presented in [1, Section  4.6] and [10]. In Section 3, we solve our main problem formulated above. At the basis of this solution is the formula𝛽_𝑘=𝑎∏^𝑁_{𝑗=1,𝑗≠𝑘}𝜆_𝑗−𝜆_𝑘∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑘, (1.4) where1𝑎=^𝑁_𝑚=11∏^𝑁_{𝑗=1,𝑗≠𝑚}𝜆_𝑗−𝜆_𝑚∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑚. (1.5) These formulae express the normalizing numbers 𝛽_𝑘 of a finite Jacobi matrix in terms of two of its spectra. The formulae (1.4) and (1.5) give a conditional solution (i.e., assuming that there exists a matrix of the form (1.1) which has the sequences {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 as two of its spectra) of the inverse problem in terms of two spectra because once we know the numbers {𝜆_𝑘}^𝑁_𝑘=1 and {𝛽_𝑘}^𝑁_𝑘=1, we can form the matrix 𝐽 by the prescription given in Section 2. Next, we give necessary and sufficient conditions for two sequences of real numbers {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 to be two spectra of a Jacobi matrix of the form (1.1) with entries in the class (1.2), that is, we solve the main problem of this paper. The conditions consist of the following single and simple condition:𝜆₁<𝜇₁<𝜆₂<𝜇₂<𝜆₃⋯<𝜆_𝑁−1<𝜇_𝑁−1<𝜆_𝑁, (1.6) that is, the numbers 𝜆_𝑘 and 𝜇_𝑘 interlace.

2. Preliminaries on the Inverse Spectral Problem

In this section, we follow the author’s paper [10]. Given a Jacobi matrix 𝐽 of the form (1.1) with the entries (1.2), consider the eigenvalue problem 𝐽𝑦=𝜆𝑦 for a column vector 𝑦={𝑦_𝑛}^𝑁−1_𝑛=0, that is equivalent to the second-order linear difference equation𝑎_𝑛−1𝑦_𝑛−1+𝑏_𝑛𝑦_𝑛+𝑎_𝑛𝑦_𝑛+1=𝜆𝑦_𝑛,𝑛∈{0,1,…,𝑁−1},𝑎₋₁=𝑎_𝑁−1=1, (2.1) for {𝑦_𝑛}^𝑁_𝑛=−1, with the boundary conditions:𝑦₋₁=𝑦_𝑁=0. (2.2) Denote by {𝑃_𝑛(𝜆)}^𝑁_𝑛=−1 and {𝑄_𝑛(𝜆)}^𝑁_𝑛=−1 the solutions of (2.1) satisfying the initial conditions:𝑃₋₁(𝜆)=0,𝑃₀(𝜆)𝑄=1,₋₁(𝜆)=−1,𝑄₀(𝜆)=0. (2.3)(2.4) For each 𝑛≥0,𝑃_𝑛(𝜆) is a polynomial of degree 𝑛 and is called a polynomial of first kind and 𝑄_𝑛(𝜆) is a polynomial of degree 𝑛−1 and is known as a polynomial of second kind. The equalitydet(𝐽−𝜆𝐼)=(−1)^𝑁𝑎₀𝑎₁⋯𝑎_𝑁−2𝑃_𝑁(𝜆) (2.5) holds so that the eigenvalues of the matrix 𝐽 coincide with the zeros of the polynomial 𝑃_𝑁(𝜆). If 𝑃_𝑁(𝜆₀)=0, then {𝑃_𝑛(𝜆₀)}^𝑁−1_𝑛=0 is an eigenvector of 𝐽 corresponding to the eigenvalue 𝜆₀. Any eigenvector of 𝐽 corresponding to the eigenvalue 𝜆₀ is a constant multiple of {𝑃_𝑛(𝜆₀)}^𝑁−1_𝑛=0.

As shown in [10, Section  8], the equations𝑃_𝑁−1(𝜆)𝑄_𝑁(𝜆)−𝑃_𝑁(𝜆)𝑄_𝑁−1(𝜆)𝑃=1,_𝑁−1(𝜆)𝑃^′_𝑁(𝜆)−𝑃_𝑁(𝜆)𝑃^′_𝑁−1(𝜆)=^𝑁−1_𝑛=0𝑃²_𝑛(𝜆) (2.6)(2.7) hold, where the prime denotes the derivative with respect to 𝜆.

Since the real Jacobi matrix 𝐽 of the form (1.1), (1.2) is self-adjoint, its eigenvalues are real. Let 𝜆₀ be a zero of the polynomial 𝑃_𝑁(𝜆). The zero 𝜆₀ is an eigenvalue of the matrix 𝐽 by (2.5), and hence it is real. Putting 𝜆=𝜆₀ in (2.7) and using 𝑃_𝑁(𝜆₀)=0, we get𝑃_𝑁−1𝜆₀𝑃^′_𝑁𝜆₀=^𝑁−1_𝑛=0𝑃²_𝑛𝜆₀. (2.8) The right-hand side of (2.8) is different from zero because the polynomials 𝑃_𝑛(𝜆) have real coefficients and hence are real for real values of 𝜆, and besides 𝑃₀(𝜆)=1. Therefore, 𝑃^′_𝑁(𝜆₀)≠0, that is, the zero 𝜆₀ of the polynomial 𝑃_𝑁(𝜆) is simple. Hence the 𝑃_𝑁(𝜆), as a polynomial of degree 𝑁, has 𝑁 distinct zeros. Thus, any real Jacobi matrix 𝐽 of the form (1.1), (1.2) has precisely 𝑁 real and distinct eigenvalues.

Let 𝑅(𝜆)=(𝐽−𝜆𝐼)⁻¹ be the resolvent of the matrix 𝐽 (by 𝐼 we denote the identity matrix of needed dimension) and 𝑒₀ the 𝑁-dimensional column vector with the components 1,0,…,0. The rational function𝑤(𝜆)=−𝑅(𝜆)𝑒₀,𝑒₀=(𝜆𝐼−𝐽)⁻¹𝑒₀,𝑒₀, (2.9) we call the resolvent function of the matrix 𝐽, where ⟨⋅,⋅⟩ stands for the standard inner product in ℂ^𝑁. This function is known also as the Weyl-Titchmarsh function of 𝐽.

In [10, Section  5] it is shown that the entries 𝑅_𝑛𝑚(𝜆) of the matrix 𝑅(𝜆)=(𝐽−𝜆𝐼)⁻¹ (resolvent of 𝐽) are of the form𝑅_𝑛𝑚(𝜆)=𝑃_𝑛(𝜆)𝑄_𝑚(𝜆)+𝑀(𝜆)𝑃_𝑚(𝜆𝑃),0≤𝑛≤𝑚≤𝑁−1,_𝑚(𝜆)𝑄_𝑛(𝜆)+𝑀(𝜆)𝑃_𝑛(𝜆),0≤𝑚≤𝑛≤𝑁−1, (2.10) where𝑀(𝜆)𝑄=−^𝑁(𝜆)𝑃_𝑁(𝜆). (2.11) Therefore, according to (2.9) and using initial conditions (2.3) and (2.4), we get𝑤(𝜆)=−𝑅₀₀(𝜆)(𝜆)=𝑄=−𝑀^𝑁(𝜆)𝑃_𝑁(𝜆). (2.12)

We often will use the following well-known simple useful lemma. We bring it here for easy reference.

Lemma 2.1. Let 𝐴(𝜆) and 𝐵(𝜆) be polynomials with complex coefficients and deg𝐴<deg𝐵. Next, suppose that 𝐵(𝜆)=𝑏(𝜆−𝑧₁)⋯(𝜆−𝑧_𝑁), where 𝑧₁,…,𝑧_𝑁 are distinct complex numbers and 𝑏 is a nonzero complex number. Then, there exist uniquely determined complex numbers 𝑎₁,…,𝑎_𝑁 such that 𝐴(𝜆)𝐵(𝜆)=^𝑁_𝑘=1𝑎_𝑘𝜆−𝑧_𝑘, (2.13) for all values of 𝜆 different from 𝑧₁,…,𝑧_𝑁. The numbers 𝑎_𝑘 are given by the equation 𝑎_𝑘=lim_{𝜆→𝑧𝑘}𝜆−𝑧_𝑘𝐴(𝜆)𝐵(𝜆)=𝐴𝑧_𝑘𝐵^′𝑧_𝑘,𝑘∈{1,…,𝑁}. (2.14)

Proof. For each 𝑘∈{1,…,𝑁}, define the polynomial 𝐿_𝑘(𝜆)=𝑏^𝑁_{𝑗=1,𝑗≠𝑘}𝜆−𝑧_𝑗=𝐵(𝜆)𝜆−𝑧_𝑘, (2.15) of degree 𝑁−1 and set 𝐹(𝜆)=𝐴(𝜆)−^𝑁_𝑘=1𝑎_𝑘𝐿_𝑘(𝜆), (2.16) where 𝑎_𝑘 is defined by (2.14). Obviously 𝐹(𝜆) is a polynomial and deg𝐹≤𝑁−1 (recall that deg𝐴<deg𝐵=𝑁). Since 𝐿_𝑘𝑧_𝑗=0for𝑗≠𝑘,𝐿_𝑘𝑧_𝑘=𝐵^′𝑧_𝑘≠0, (2.17) we have 𝐹𝑧_𝑗=𝐴𝑧_𝑗−^𝑁_𝑘=1𝑎_𝑘𝐿_𝑘𝑧_𝑗=𝐴𝑧_𝑗−𝑎_𝑗𝐿_𝑗𝑧_𝑗=𝐴𝑧_𝑗−𝐴𝑧_𝑗𝐵^′𝑧_𝑗𝐵^′𝑧_𝑗=0, (2.18) for all 𝑗∈{1,…,𝑁}. Thus, the polynomial 𝐹(𝜆) of degree ≤𝑁−1 has 𝑁 distinct zeros 𝑧₁,…,𝑧_𝑁. Then 𝐹(𝜆)≡0 and we get 𝐴(𝜆)=^𝑁_𝑘=1𝑎_𝑘𝐿_𝑘(𝜆)=^𝑁_𝑘=1𝑎_𝑘𝐵(𝜆)𝜆−𝑧_𝑘=𝐵(𝜆)^𝑁_𝑘=1𝑎_𝑘𝜆−𝑧_𝑘. (2.19) This proves (2.13). Note that the decomposition (2.13) is unique as for the 𝑎_𝑘 in this decomposition (2.14) necessarily holds.

Denote by 𝜆₁,…,𝜆_𝑁 all the zeros of the polynomial 𝑃_𝑁(𝜆) (which coincide by (2.5) with the eigenvalues of the matrix 𝐽 and which are real and distinct):𝑃_𝑁(𝜆)=𝑐𝜆−𝜆₁⋯𝜆−𝜆_𝑁, (2.20) where 𝑐 is a nonzero constant. Therefore applying Lemma 2.1 to (2.12), we can get for the resolvent function 𝑤(𝜆) the following decomposition:𝑤(𝜆)=^𝑁_𝑘=1𝛽_𝑘𝜆−𝜆_𝑘, (2.21) where𝛽_𝑘=𝑄_𝑁𝜆_𝑘𝑃^′_𝑁𝜆_𝑘. (2.22) Further, putting 𝜆=𝜆_𝑘 in (2.6) and (2.7) and taking into account that 𝑃_𝑁(𝜆_𝑘)=0, we get𝑃_𝑁−1𝜆_𝑘𝑄_𝑁𝜆_𝑘𝑃=1,_𝑁−1𝜆_𝑘𝑃^′_𝑁𝜆_𝑘=^𝑁−1_𝑛=0𝑃²_𝑛𝜆_𝑘, (2.23)(2.24) respectively. It follows from (2.23) that 𝑄_𝑁(𝜆_𝑘)≠0 and therefore 𝛽_𝑘≠0. Comparing (2.22), (2.23), and (2.24), we find that𝛽_𝑘=^𝑁−1_𝑛=0𝑃²_𝑛𝜆_𝑘⁻¹, (2.25) whence we obtain, in particular, that 𝛽_𝑘>0.

Since {𝑃_𝑛(𝜆_𝑘)}^𝑁−1_𝑛=0 is an eigenvector of the matrix 𝐽 corresponding to the eigenvalue 𝜆_𝑘, it is natural, according to the formula (2.25), to call 𝛽_𝑘 the normalizing number of the matrix 𝐽 corresponding to the eigenvalue 𝜆_𝑘.

The collection of the eigenvalues and normalizing numbers:𝜆_𝑘,𝛽_𝑘(𝑘=1,…,𝑁), (2.26) of the matrix 𝐽 of the form (1.1), (1.2) is called the spectral data of this matrix.

Determination of the spectral data of a given Jacobi matrix is called the direct spectral problem for this matrix.

Thus, the spectral data consist of the eigenvalues and associated normalizing numbers derived by decomposing the resolvent function (Weyl-Titchmarsh function) into partial fractions using the eigenvalues. The resolvent function 𝑤(𝜆) of the matrix 𝐽 can be constructed by using (2.12). Another convenient formula for computing the resolvent function is (see [10, Section  5])𝑤(𝜆)𝐽=−det⁽¹⁾−𝜆𝐼(),det𝐽−𝜆𝐼 (2.27) where 𝐽⁽¹⁾ is the matrix obtained from 𝐽 by deleting the first row and first column of 𝐽.

It follows from (2.27) that 𝜆𝑤(𝜆) tends to 1 as 𝜆→∞. Therefore multiplying (2.21) by 𝜆 and passing then to the limit as 𝜆→∞, we find^𝑁_𝑘=1𝛽_𝑘=1. (2.28)

The inverse spectral problem is stated as follows.(i)To see if it is possible to reconstruct the matrix 𝐽, given its spectral data (2.26). If it is possible, to describe the reconstruction procedure;(ii)To find the necessary and sufficient conditions for a given collection (2.26) to be spectral data for some matrix 𝐽 of the form (1.1) with entries belonging to the class (1.2).

The solution of this problem is well known (see [1, Section  4.6] and [10]) and let us bring here the final result.

Given a collection (2.26), where 𝜆₁,…,𝜆_𝑁 are real and distinct and 𝛽₁,…,𝛽_𝑁 are positive, define the numbers:𝑠_𝑙=^𝑁_𝑘=1𝛽_𝑘𝜆^𝑙_𝑘,𝑙=0,1,2,…, (2.29) and using these numbers introduce the determinants:𝐷_𝑛=𝑠₀𝑠₁⋯𝑠_𝑛𝑠₁𝑠₂⋯𝑠_𝑛+1𝑠⋮⋮⋱⋮_𝑛𝑠_𝑛+1⋯𝑠_2𝑛,𝑛=0,1,2,…. (2.30)

Lemma 2.2. For the determinants 𝐷_𝑛 defined by (2.30) and (2.29), we have 𝐷_𝑛>0 for 𝑛∈{0,1,…,𝑁−1} and 𝐷_𝑛=0 for 𝑛≥𝑁.

Proof. Denote by 𝐴 the (𝑛+1)×(𝑛+1) matrix corresponding to the determinant 𝐷_𝑛 given by (2.30). Then for arbitrary real column vector 𝑥=(𝑥₀,𝑥₁,…,𝑥_𝑛)^𝑇, we have ⟨𝐴𝑥,𝑥⟩=^𝑛_𝑗,𝑚=0𝑠_𝑗+𝑚𝑥_𝑚𝑥_𝑗=^𝑛_𝑗,𝑚=0^𝑁_𝑘=1𝛽_𝑘𝜆_𝑘^𝑗+𝑚𝑥_𝑚𝑥_𝑗=^𝑁_𝑘=1𝛽_𝑘^𝑛_𝑗=0𝑥_𝑗𝜆^𝑗_𝑘²=^𝑁_𝑘=1𝛽_𝑘𝐺𝜆_𝑘²≥0, (2.31) where 𝐺(𝜆)=^𝑛_𝑗=0𝑥_𝑗𝜆^𝑗. (2.32) Further, it follows that if ⟨𝐴𝑥,𝑥⟩=0, then 𝐺𝜆_𝑘=0,𝑘=1,…,𝑁. (2.33) If 𝑛≤𝑁−1, then deg𝐺≤𝑁−1 and (2.33) is possible only if 𝐺(𝜆)≡0 (recall that 𝜆₁,…,𝜆_𝑁 are distinct). But then 𝑥₀=𝑥₁=⋯=𝑥_𝑛=0. Therefore, ⟨𝐴𝑥,𝑥⟩>0, (2.34) for all nonzero real vectors 𝑥=(𝑥₀,𝑥₁,…,𝑥_𝑛)^𝑇 if 𝑛≤𝑁−1. Then as is well known from Linear Algebra, we have det𝐴>0. Thus we have proved that 𝐷_𝑛>0 for 𝑛≤𝑁−1.
To prove that 𝐷_𝑛=0 for 𝑛≥𝑁, let us define the linear functional Ω on the linear space of all polynomials in 𝜆 with complex coefficients as follows: if 𝐺(𝜆) is a polynomial, then the value ⟨Ω,𝐺(𝜆)⟩ of the functional Ω on the element (polynomial) 𝐺 is ⟨Ω,𝐺(𝜆)⟩=^𝑁_𝑘=1𝛽_𝑘𝐺𝜆_𝑘. (2.35) Let 𝑚≥0 be a fixed integer and set 𝑇(𝜆)=𝜆^𝑚𝜆−𝜆₁⋯𝜆−𝜆_𝑁=𝑡_𝑚𝜆^𝑚+𝑡_𝑚+1𝜆^𝑚+1+⋯+𝑡_{𝑚+𝑁−1}𝜆^{𝑚+𝑁−1}+𝜆^𝑚+𝑁. (2.36) Then, according to (2.35), Ω,𝜆^𝑙𝑇(𝜆)=0,𝑙=0,1,2,…. (2.37) Consider (2.37) for 𝑙=0,1,2,…,𝑁+𝑚, and substitute (2.36) in it for 𝑇(𝜆). Taking into account that Ω,𝜆^𝑙=^𝑁_𝑘=1𝛽_𝑘𝜆^𝑙_𝑘=𝑠_𝑙,𝑙=0,1,2,…, (2.38) we get 𝑡_𝑚𝑠_𝑙+𝑚+𝑡_𝑚+1𝑠_𝑙+𝑚+1+⋯+𝑡_{𝑚+𝑁−1}𝑠_{𝑙+𝑚+𝑁−1}+𝑠_{𝑙+𝑚+𝑁}=0,𝑙=0,1,2,…,𝑁+𝑚. (2.39) Therefore, (0,…,0,𝑡_𝑚,𝑡_𝑚+1,…,𝑡_{𝑚+𝑁−1},1) is a nontrivial solution of the homogeneous system of linear algebraic equations: 𝑥₀𝑠_𝑙+𝑥₁𝑠_𝑙+1+⋯+𝑥_𝑚𝑠_𝑙+𝑚+𝑥_𝑚+1𝑠_𝑙+𝑚+1+⋯+𝑥_{𝑚+𝑁−1}𝑠_{𝑙+𝑚+𝑁−1}+𝑥_𝑚+𝑁𝑠_{𝑙+𝑚+𝑁}=0,𝑙=0,1,2,…,𝑁+𝑚, (2.40) with the unknowns 𝑥₀,𝑥₁,…,𝑥_𝑚,𝑥_𝑚+1,…,𝑥_{𝑚+𝑁−1},𝑥_𝑚+𝑁. Therefore, the determinant of this system, which coincides with 𝐷_𝑁+𝑚, must be equal to zero.

Theorem 2.3. Let an arbitrary collection (2.26) of numbers be given. In order for this collection to be the spectral data for a Jacobi matrix 𝐽 of the form (1.1) with entries belonging to the class (1.2), it is necessary and sufficient that the following two conditions be satisfied:(i)The numbers 𝜆₁,…,𝜆_𝑁 are real and distinct.(ii)The numbers 𝛽₁,…,𝛽_𝑁 are positive and such that 𝛽₁+⋯+𝛽_𝑁=1.
Under the conditions (i) and (ii) we have 𝐷_𝑛>0 for 𝑛∈{0,1,…,𝑁−1} and the entries 𝑎_𝑛 and 𝑏_𝑛 of the matrix 𝐽 for which the collection (2.26) is spectral data, are recovered by the formulae 𝑎_𝑛=±𝐷_𝑛−1𝐷_𝑛+1𝐷_𝑛,𝑛∈{0,1,…,𝑁−2},𝐷₋₁𝑏=1,_𝑛=Δ_𝑛𝐷_𝑛−Δ_𝑛−1𝐷_𝑛−1,𝑛∈{0,1,…,𝑁−1},Δ₋₁=0,Δ₀=𝑠₁, (2.41)(2.42) where 𝐷_𝑛 is defined by (2.30) and (2.29), and Δ_𝑛 is the determinant obtained from the determinant 𝐷_𝑛 by replacing in 𝐷_𝑛 the last column by the column with the components 𝑠_𝑛+1,𝑠_𝑛+2,…,𝑠_2𝑛+1.

It follows from the above solution of the inverse problem that the matrix (1.1) is not uniquely restored from the spectral data. This is linked with the fact that the 𝑎_𝑛 are determined from (2.41) uniquely up to a sign. To ensure that the inverse problem is uniquely solvable, we have to specify additionally a sequence of signs + and −. Namely, let {𝜎₀,𝜎₁,…,𝜎_𝑁−2} be a given finite sequence, where for each 𝑛∈{0,1,…,𝑁−2} the 𝜎_𝑛 is + or −. We have 2^𝑁−1 such different sequences. Now to determine 𝑎_𝑛 uniquely from (2.41) for 𝑛∈{0,1,…,𝑁−2}, we can choose the sign 𝜎_𝑛 when extracting the square root. In this way, we get precisely 2^𝑁−1 distinct Jacobi matrices possessing the same spectral data. The inverse problem is solved uniquely from the data consisting of the spectral data and a sequence {𝜎₀,𝜎₁,…,𝜎_𝑁−2} of signs + and −. Thus, we can say that the inverse problem with respect to the spectral data is solved uniquely up to signs of the off-diagonal elements of the recovered Jacobi matrix. In particular, the inverse problem is solvable uniquely in the class of entries 𝑎_𝑛>0,  𝑏_𝑛∈ℝ.

3. Inverse Problem for Two Spectra

Let 𝐽 be an 𝑁×𝑁 Jacobi matrix of the form (1.1) with entries satisfying (1.2). Define 𝐽₁ to be the truncated Jacobi matrix given by (1.3). We denote the eigenvalues of the matrices 𝐽 and 𝐽₁ by 𝜆₁<⋯<𝜆_𝑁 and 𝜇₁<⋯<𝜇_𝑁−1, respectively. We call the collections {𝜆_𝑘(𝑘=1,…,𝑁)} and {𝜇_𝑘(𝑘=1,…,𝑁−1)} the two spectra of the matrix 𝐽.

The inverse problem for two spectra consists in the reconstruction of the matrix 𝐽 by two of its spectra.

We will reduce the inverse problem for two spectra to the inverse problem for eigenvalues and normalizing numbers solved above in Section 2.

First, let us study some necessary properties of the two spectra of the Jacobi matrix 𝐽.

Let 𝑃_𝑛(𝜆) and 𝑄_𝑛(𝜆) be the polynomials of the first and second kind for the matrix 𝐽. By (2.5) we havedet(𝐽−𝜆𝐼)=(−1)^𝑁𝑎₀𝑎₁⋯𝑎_𝑁−2𝑃_𝑁(𝜆),det𝐽₁−𝜆𝐼=(−1)^𝑁−1𝑎₀𝑎₁⋯𝑎_𝑁−2𝑃_𝑁−1(𝜆). (3.1)(3.2) Note that we have used the fact that 𝑎_𝑁−1=1. Therefore, the eigenvalues 𝜆₁,…,𝜆_𝑁 and 𝜇₁,…,𝜇_𝑁−1 of the matrices 𝐽 and 𝐽₁ coincide with the zeros of the polynymials 𝑃_𝑁(𝜆) and 𝑃_𝑁−1(𝜆), respectively.

Dividing both sides of (2.6) by 𝑃_𝑁−1(𝜆)𝑃_𝑁(𝜆) gives𝑄_𝑁(𝜆)𝑃_𝑁(𝜆)−𝑄_𝑁−1(𝜆)𝑃_𝑁−1(𝜆)=1𝑃_𝑁−1(𝜆)𝑃_𝑁(𝜆). (3.3) Therefore, by formula (2.12) for the resolvent function 𝑤(𝜆), we obtain𝑤(𝜆)=𝑄_𝑁−1(𝜆)𝑃_𝑁−1(𝜆)+1𝑃_𝑁−1(𝜆)𝑃_𝑁(𝜆). (3.4)

Lemma 3.1. The matrices 𝐽 and 𝐽₁ have no common eigenvalues, that is, 𝜆_𝑘≠𝜇_𝑗 for all values of 𝑘 and 𝑗.

Proof. Suppose that 𝜆 is an eigenvalue of the matrices 𝐽 and 𝐽₁. Then by (3.1) and (3.2) we have 𝑃_𝑁(𝜆)=𝑃_𝑁−1(𝜆)=0. But this is impossible by (2.6).

Lemma 3.2. The equality (trace formula) 𝜆_𝑁+^𝑁−1_𝑘=1𝜆_𝑘−𝜇_𝑘=𝑏_𝑁−1 (3.5) holds.

Proof. For any matrix 𝐴=[𝑎_𝑗𝑘]^𝑁_𝑗,𝑘=1 the spectral trace of 𝐴 coincides with the matrix trace of 𝐴∶ If 𝜈₁,…,𝜈_𝑁 are the eigenvalues of 𝐴, then ^𝑁_𝑘=1𝜈_𝑘=^𝑁_𝑘=1𝑎_𝑘𝑘. (3.6) Therefore, we can write ^𝑁_𝑘=1𝜆_𝑘=𝑏₀+𝑏₁+⋯+𝑏_𝑁−2+𝑏_𝑁−1,^𝑁−1_𝑘=1𝜇_𝑘=𝑏₀+𝑏₁+⋯+𝑏_𝑁−2. (3.7) Subtracting the last two equalities side by side, we arrive at (3.5).

Lemma 3.3. The eigenvalues of 𝐽 and 𝐽₁ interlace: 𝜆₁<𝜇₁<𝜆₂<𝜇₂<𝜆₃⋯<𝜆_𝑁−1<𝜇_𝑁−1<𝜆_𝑁. (3.8)

Proof. Let us set 𝜓(𝜆)=𝑃_𝑁−1(𝜆)𝑃_𝑁(𝜆), (3.9) so that 𝜓(𝜆) is a rational function whose poles coincide with the eigenvalues of 𝐽 and whose zeros coincide with the eigenvalues of 𝐽₁. Applying Lemma 2.1 to the rational function 𝜓(𝜆) we can write 𝜓(𝜆)=^𝑁_𝑘=1𝛾_𝑘𝜆−𝜆_𝑘, (3.10) where 𝛾_𝑘=𝑃_𝑁−1𝜆_𝑘𝑃^′_𝑁𝜆_𝑘. (3.11) Next, (2.24) shows that 𝑃_𝑁−1(𝜆_𝑘)𝑃^′_𝑁(𝜆_𝑘)>0, that is, 𝑃_𝑁−1(𝜆_𝑘) and 𝑃^′_𝑁(𝜆_𝑘) have the same sign. Then (3.11) implies that 𝛾_𝑘>0(𝑘=1,…,𝑁).
Differentiating (3.10) we get 𝜓^′(𝜆)=−^𝑁_𝑘=1𝛾_𝑘𝜆−𝜆_𝑘². (3.12) It follows from (3.12) that 𝜓^′(𝜆)<0 for real values of 𝜆, different from 𝜆₁,…,𝜆_𝑁. Therefore, 𝜓(𝜆) is strictly decreasing continuous function on the intervals (−∞,𝜆₁),  (𝜆₁,𝜆₂),…,(𝜆_𝑁−1,𝜆_𝑁),  (𝜆_𝑁,∞). Besides, it follows from (3.10) that lim_|𝜆|→∞𝜓(𝜆)=0,lim_{𝜆→𝜆−𝑘}𝜓(𝜆)=−∞,lim_{𝜆→𝜆+𝑘}𝜓(𝜆)=∞. (3.13) Consequently, the function 𝜓(𝜆) has no zero in the intervals (−∞,𝜆₁) and (𝜆_𝑁,∞), and exactly one zero in each of the intervals (𝜆₁,𝜆₂),…,(𝜆_𝑁−1,𝜆_𝑁). Since the zeros of the function 𝜓(𝜆) coincide with the eigenvalues of 𝐽₁ by (3.9), the proof is complete.

The following lemma gives a formula for calculating the normalizing numbers 𝛽₁,…,𝛽_𝑁 in terms of the two spectra.

Lemma 3.4. For each 𝑘∈{1,…,𝑁} the formula 𝛽_𝑘=𝑎∏^𝑁_{𝑗=1,𝑗≠𝑘}𝜆_𝑗−𝜆_𝑘∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑘 (3.14) holds, where 1𝑎=^𝑁_𝑚=11∏^𝑁_{𝑗=1,𝑗≠𝑚}𝜆_𝑗−𝜆_𝑚∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑚. (3.15)

Proof. Substituting (2.21) in the left-hand side of (3.4), we can write ^𝑁_𝑚=1𝛽_𝑚𝜆−𝜆_𝑚=𝑄_𝑁−1(𝜆)𝑃_𝑁−1(𝜆)+1𝑃_𝑁−1(𝜆)𝑃_𝑁(𝜆). (3.16) Multiply both sides of the last equality by 𝜆−𝜆_𝑘 and pass then to the limit as 𝜆→𝜆_𝑘. Taking into account that 𝑃_𝑁(𝜆_𝑘)=0,  𝑃^′_𝑁(𝜆_𝑘)≠0,  𝑃_𝑁−1(𝜆_𝑘)≠0 (see (2.23) and (2.24)), we get 𝛽_𝑘=1𝑃^′_𝑁𝜆_𝑘𝑃_𝑁−1𝜆_𝑘. (3.17) Next, by (3.1) and (3.2) we have (−1)^𝑁𝑎₀𝑎₁⋯𝑎_𝑁−2𝑃_𝑁(𝜆)=^𝑁_𝑗=1𝜆_𝑗,()−𝜆−1^𝑁−1𝑎₀𝑎₁⋯𝑎_𝑁−2𝑃_𝑁−1(𝜆)=∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆. (3.18) Substituting these in the right-hand side of (3.17), we obtain 𝛽_𝑘=𝑎∏^𝑁_{𝑗=1,𝑗≠𝑘}𝜆_𝑗−𝜆_𝑘∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑘, (3.19) where 𝑎=𝑎₀𝑎₁⋯𝑎_𝑁−2². (3.20) Replacing 𝑘 by 𝑚 in (3.19) and then summing this equation over 𝑚=1,…,𝑁 and taking into account (2.28), we get (3.15). The lemma is proved.

Theorem 3.5. (uniqueness result). The two spectra {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 of the Jacobi matrix 𝐽 of the form (1.1) in the class 𝑎_𝑛>0,𝑏_𝑛∈ℝ, (3.21) uniquely determine the matrix 𝐽.

Proof. Given the two spectra {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 of the matrix 𝐽, we determine uniquely the normalizing numbers 𝛽_𝑘(𝑘=1,…,𝑁) of the matrix 𝐽 by (3.14) and (3.15). Since the collection of the eigenvalues and normalizing numbers {𝜆_𝑘,𝛽_𝑘(𝑘=1,…,𝑁)} of the matrix 𝐽 determines 𝐽 uniquely in the class (3.21), the proof is complete.

The following theorem solves the inverse problem in terms of the two spectra. Its proof given below contains an effective procedure for the construction of the Jacobi matrix from its two spectra.

Theorem 3.6. In order for giving two collections of real numbers {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 to be the spectra of two matrices 𝐽 and 𝐽₁, respectively, of the forms (1.1) and (1.3) with the entries in the class (1.2), it is necessary and sufficient that the following inequalities be satisfied: 𝜆₁<𝜇₁<𝜆₂<𝜇₂<𝜆₃⋯<𝜆_𝑁−1<𝜇_𝑁−1<𝜆_𝑁. (3.22)

Proof. The necessity of the condition (3.22) has been proved above in Lemma 3.3. To prove the sufficiency, suppose that two collections of real numbers {𝜆_𝑘}^𝑁_𝑘=1 and {𝜇_𝑘}^𝑁−1_𝑘=1 are given which satisfy the inequalities in (3.22). We construct 𝛽_𝑘(𝑘=1,…,𝑁) according to these data by (3.14) and (3.15). It follows from (3.22) that ^𝑁_{𝑗=1,s𝑗≠𝑘}𝜆_𝑗−𝜆_𝑘^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑘>0,𝑘=1,…,𝑁. (3.23) Therefore, the expression on the right-hand side of (3.14) is positive and hence 𝛽_𝑘>0(𝑘=1,…,𝑁). Next, it follows directly from (3.14) and (3.15) that 𝛽₁+⋯+𝛽_𝑁=1.
Consequently, the collection {𝜆_𝑘,𝛽_𝑘(𝑘=1,…,𝑁)} satisfies the conditions of Theorem 2.3, and hence there exists a Jacobi matrix 𝐽 of the form (1.1) with entries from the class (1.2) such that the 𝜆_𝑘(𝑘=1,…,𝑁) are the eigenvalues and the 𝛽_𝑘(𝑘=1,…,𝑁) are the corresponding normalizing numbers for 𝐽. Having the matrix 𝐽, we construct the matrix 𝐽₁ by (1.3). It remains to show that {𝜇_𝑘}^𝑁−1_𝑘=1 is the spectrum of the constructed matrix 𝐽₁. Denote the eigenvalues of 𝐽₁ by 𝜇₁<⋯<𝜇_𝑁−1. By Lemma 3.3, 𝜆₁<𝜇₁<𝜆₂<𝜇₂<⋯<𝜆_𝑁−1<𝜇_𝑁−1<𝜆_𝑁. (3.24) We have to show that 𝜇_𝑘=𝜇_𝑘(𝑘=1,…,𝑁−1).
By the direct spectral problem, we have (Lemma 3.4): 𝛽_𝑘=𝑎∏^𝑁_{𝑗=1,𝑗≠𝑘}𝜆_𝑗−𝜆_𝑘∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑘, (3.25) where 1=𝑎^𝑁_𝑚=11∏^𝑁_{𝑗=1,𝑗≠𝑚}𝜆_𝑗−𝜆_𝑚∏^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑚. (3.26) On the other hand, by our construction of 𝛽_𝑘, we have (3.14) and (3.15). Equating the right-hand sides of (3.25) and (3.14), we obtain 𝑎^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑘=𝑎^𝑁−1_𝑗=1𝜇_𝑗−𝜆_𝑘,𝑘=1,…,𝑁. (3.27) This means that the polynomial 𝑎^𝑁−1_𝑗=1𝜆−𝜇_𝑗−𝑎^𝑁−1_𝑗=1𝜆−𝜇_𝑗, (3.28) of degree ≤𝑁−1 has 𝑁 distinct zeros 𝜆₁,…,𝜆_𝑁. Then, this polynomial identically equals zero. Hence, 𝑎=𝑎 and 𝜇_𝑗=𝜇_𝑗(𝑗=1,…,𝑁). The proof is complete.