Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator

Theorem 1.1

Let { γ i } i = 1 ∞ , { λ i } i = 1 ∞ be the eigenvalues of the operators L 0 , L , respectively, then c ( x ) in (1.1) can be recovered by { γ i } i = 1 ∞ , { λ i } i = 1 ∞ .

Proposition 2.1

Let { γ i } i = 1 ∞ , { λ i } i = 1 ∞ be the eigenvalues of the operators L 0 , L , respectively, then the following relation holds:

or

Proof

By (2.1) and [2, p. 15],

where G ( x , y , z ) is the Green’s function for L 0 and c i = ∫ 0 1 c ( x ) v i ( x ) d x . Thus, F ( z ) is meromorphic with simple poles in the points z = γ n i , n i ∈ { j ∈ N + ∣ c j ≠ 0 } . Moreover, for x ∈ R \ { γ n i } i = 1 ∞ , it follows

and

Let { a i } i = 1 ∞ be the zero set of α F ( x + i 0 ) + 1 . In the case α = 1 , by (2.6) and (2.7),

and

Then there is no zero of F ( x + i 0 ) + 1 in ( − ∞ , γ n 1 ) and exactly one zero in ( γ n i , γ n i + 1 ) , that is,

All zeros of F ( z ) + 1 are the eigenvalues of the operator L , thus the zeros of F ( z ) + 1 are real and consist of the zeros of F ( x + i 0 ) . Then we have the multiplicity function

Using (2.3), it follows

Combining with (2.11), we have (2.4). Similarly, in the case α = − 1 , (2.5) holds. The proof is complete.□

Proof of Theorem 1.1

By (2.12), there is

Let

In the case α = 1 , by (2.8)–(2.10), we have

and

then

Similarly, in the case α = − 1 ,

By (I.16) and (I.19) in [1],

that is,

Combining with (2.6), ∣ c i ∣ 2 = lim z → γ i ( γ i − z ) F ( z ) . Thus, we can construct

where v i ( x ) is the normalized eigenfunction corresponding to the eigenvalue γ i , and the proof is therefore complete.□

Algorithm 2.2

1. From the given sequences { γ n , λ n } n = 1 ∞ , we construct the functions F ( z ) by (2.13) and (2.14);

2. Find the number ∣ c i ∣ by ∣ c i ∣ 2 = lim z → λ i ( λ i − z ) F ( z ) ;

3. Calculate c ( x ) by (2.15).