Inverse Eigenvalue Problems for Singular Rank One Perturbations of a Sturm-Liouville Operator

This paper is concerned with the inverse eigenvalue problem for singular rank one perturbations of a Sturm-Liouville operator. We determine uniquely the potential function from the spectra of the Sturm-Liouville operator and its rank one perturbations.


Introduction
Consider the boundary problem on ð0, 1Þ with where qðxÞ ∈ L 2 ð0, 1Þ is real valued, α ∈ ℝ \ f0g, x 0 ∈ ð0, 1Þ, and δðxÞ are the Dirac delta function. It is well known [1] that the operatorL is a self-adjoint operator in L 2 ð0, 1Þ, which is a singular rank one perturbation of the Sturm-Liouville operator Ly ≔ −y′′ + qðxÞy. The goal of this paper is to deal with the inverse problem of recovering the potential qðxÞ in (1) from the spectra of L andL, by applying the method in [1] and the perturbation theory for linear operators [2]. Note that in [3], the boundary problem with (2) and the discontinuous conditions can be regarded as the problem (1)- (2). The research of this paper can be a variation of Borg's two-spectra theorem [4] for the second spectrum that is obtained by attaching the interface conditions (4) and (5) to the problem (3) and (2). Our immediate motivation is a recent research of del Rio and Kudryavtsev [5,6], who considered the inverse problem for Jacobi matrices and recovered the original matrix from the spectra of it and its interior mass-spring perturbation; they indicated that the uniqueness for the inverse problem does not remain valid in general; however, under certain conditions, there exists at most a finite number of matrices corresponding to the two spectra. Such operatorL appears not only in electronics but in other areas such as the theory of diffusion processes, see the related references in [7,8]. Some spectral and inverse spectral problems for the Sturm-Liouville operator with rank one perturbations have been investigated in [1,3,[9][10][11][12][13][14][15][16][17][18][19][20]. In particular, Albeverio, Hryniv, and Nizhnik [3] considered the inverse eigenvalue problem for the Sturm-Liouville operator with the point potential vðxÞðy, δÞ L 2 and the perturbation δ ðx − x 0 Þðy, vÞ L 2 where vðxÞ ∈ L 2 ð0, 1Þ and x 0 = 1. Later, Nizhnik [16] continued the problem with x 0 ∈ ð0, 1Þ. However, we consider the inverse problem for the operator (1) with the potential qðxÞ ∈ L 2 ð0, 1Þ and the perturbation δðx − x 0 Þ ðy, δÞ L 2 . And the potential may not be determined uniquely just by the spectra; so, we employ the addition information. Moreover, the approach we use can also solve the problem with the perturbation cðxÞðy, cÞ L 2 , cðxÞ ∈ L 2 ð0, 1Þ in the research [18]. In the paper, we establish the expression of the characteristic function ofL which provides a necessary preliminary for treating with its inverse eigenvalue problem. The approach we use to prove our results can convert the problem (1) and (2) into three spectra inverse problem in [21,22]. Actually, the spectra of L andL may not determine the potential uniquely (see Remark below for details). The key difficulty encountered is to identify the eigenvalues of two Sturm-Liouville problems defined on ½0, x 0 and ½x 0 , 1 from the knowledge of the spectra of L andL, for which we have to employ addition information of the number of zeros of the eigenfunctions, and the condition two spectra are disjoint.
The main result asserts that, if the spectra of L andL are disjoint, the potential qðxÞ can be determined uniquely by the spectra of L andL and the numbers of zeros, contained in ð 0, x 0 Þ, of all eigenfunctions of L. We will state and prove it in the next section.

The Main Theorem and Proof
We describe some preliminaries which will be needed subsequently, due to [1]. One defines the scale of spaces H ±1 ðLÞ associated to L as follows. The space H +1 ðLÞ is in which H +1 is easily seen to be complete. For H −1 ðLÞ, take L 2 ð0, 1Þ with the norm given by and complete it. Note that H +1 ðLÞ and H −1 ðLÞ are dual in such a way that φ ∈ H −1 is associated to the function y ∈ H +1 given by Ð 1 0 φðxÞyðxÞdx. A Sobolev estimate shows that, for any y ∈ H +1 ðLÞ, The following preliminaries are due to ( [2], p. 245-250). The multiplicity index for αFðzÞ + 1 is given by 0, for all other ζ ∈ ℂ: The multiplicity function for a closed operator T is defined bỹ +∞,for all other ζ ∈ ℂ: Let λ n be the eigenvalue of L, n ≥ 0. Then, Using the same way as the proof of the W-A formulas in ( [2], p. 248), by (10) In the following lemma, we give the spectrum and a characteristic function ofL. Lemma 1. The spectrum ofL consists of real eigenvalues. The characteristic function ofL is where ΔðλÞ is the characteristic function of L.
Then, x 0 ∈ fμ n g ∞ n=0 . For we also need to prove that x 0 is an eigenvalue ofL of multiplicity 2 if and only if x 0 is a zero of FðλÞ of multiplicity 2. Actually, if x 0 is an eigenvalue ofL of multiplicity 2, then by (21), there is x 0 ∈ fa i g ∞ i=0 ∩ fλ n g ∞ n=0 , and it is easily to get Now, we suppose that x 0 is a zero of FðλÞ of multiplicity 2, i.e., (26) holds. ByΔðx 0 Þ = 0, there is Δðx 0 Þ = 0 or αFðx 0 + i0Þ + 1 = 0. Combining with there is Δðx 0 Þ = 0 and αFðx 0 + i0Þ Then, x 0 is an eigenvalue ofL of multiplicity 2. The proof is complete. ☐ The main result in this paper is as follows.
Theorem 2. Let fλ n g ∞ n=0 and fμ n g ∞ n=0 be the spectrum of L and L, respectively. If fλ n g ∞ n=0 ∩ fμ n g ∞ n=0 = ∅, the potential qðxÞ can be determined uniquely by fλ n g ∞ n=0 , fμ n g ∞ n=0 and the numbers of zeros, contained in ð0, x 0 Þ, of all eigenfunctions of L.
Next, we prove that fλ n g ∞ n=0 , fγ − n g ∞ n=0 , and fγ + n g ∞ n=0 determine qðxÞ. By (35) and Theorem 3.2 in [21], we see that fλ n g ∞ n=0 , fγ − n g ∞ n=0 , and fγ + n g ∞ n=0 can uniquely determine qðxÞ, a.e., on ð0, 1Þ. The proof is therefore complete. ☐ Remark 3. Based on the proof of Theorem 2, we know that two disjoint spectra fλ n g ∞ n=0 and fμ n g ∞ n=0 determine fγ n g ∞ n=0 uniquely rather than fγ − n g ∞ n=0 and fγ + n g ∞ n=0 . Thus, in order to obtain the uniqueness of qðxÞ, we need to identify fγ − n g ∞ n=0 and fγ + n g ∞ n=0 from fγ n g ∞ n=0 . To this end, we have to employ the number of zeros of eigenfunctions, and the condition two spectra are disjoint, if not, we may not guarantee the uniqueness of qðxÞ. For example, if x 0 = 1/2, then the eigenvalues of L and L ± have the following asymptotic expressions: where w = 1/2 Ð 1 0 qðxÞdx, w − = Ð 1/2 0 qðxÞdx, w + = Ð 1 1/2 qðxÞdx and fκ n g, fκ − n g, fκ + n g ∈ l 2 . It is easy to see that there exists a positive integer N such that, for n > N, while we cannot identify fγ − n g ∞ n=0 and fγ + n g ∞ n=0 from fγ n g ∞ n=0 for n ≤ N, which means that there will be at most a finite number of qðxÞ corresponding to two spectra fγ n g ∞ n=0 and fλ n g ∞ n=0 in virtue of Theorem 3.2 in [21]. Hence, we employ the number of zeros of eigenfunctions, and the condition two spectra are disjoint to guarantee the uniqueness of qðxÞ. And we have not found an example that the joint spectra determine the potential uniquely.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.

Authors' Contributions
The author conceived of the study, drafted the manuscript, and approved the final manuscript.