On solvability of three spectra problem

Abstract

Three spectral problems generated by the same Sturm–Liouville equation are considered: Neumann–Dirichlet problem (the Neumann condition at the left end and the Dirichlet condition at the right end) on the whole interval [0, a], Neumann–Dirichlet problem on [0,a/2] and Dirichlet–Dirichlet problem on [a/2,a]. The three spectra inverse problem, i.e. the problem of recovering the Sturm–Liuville equation using the three spectra of these boundary value problems is completely solved.

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

“…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.…”

“…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.…”

“…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).…”

“…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…”

An inverse three spectra problem for Sturm–Liouville operators

“…In [4], the authors further generalized the uniqueness theorem by σ (0, 1; h 0 , h 1 ; q), σ (0, c; h 0 , h c ; q) and σ (c, 1; h c , h 1 ; q) with c ∈ (0, 1) and h c ∈ R ∪ {∞}. In the past years, the inverse three spectra problem has been investigated by several authors (see [5][6][7][8][9][10][11][12][13][14][15] and the references therein). The known results contain the existence, the uniqueness, the numerical scheme, the reconstructing formula, and so on.…”