A vectorial inverse nodal problem

Abstract

Abstract. Consider the vectorial Sturm-Liouville problem:is a continuous symmetric matrix-valued function defined on [0, 1], and A and B are d×d real symmetric matrices. An eigenfunction y(x) of the above problem is said to be of type (CZ) if any isolated zero of its component is a nodal point of y(x). We show that when d = 2, there are infinitely many eigenfunctions of type (CZ) if and only if (P (x), A, B) are simultaneously diagonalizable. This indicates that (P (x), A, B) can be reconstructed when all except a finite number of eigenfunctions are of type (CZ). The results supplement a theorem proved by Shen-Shieh (the second author) for Dirichlet boundary conditions. The proof depends on an eigenvalue estimate, which seems to be of independent interest.

On Inverse Nodal Problem and Multiplicities of Eigenvalues of a Vectorial Sturm-Liouville Problem

“…It was McLaughlin who initiated the study of the inverse nodal problem [9]. In the recent years, the inverse nodal problem of the Sturm-Liouville problem has been investigated a lot (see the monographs [10][11][12][13][14][15][16][17] and the references therein) and the vectorial inverse nodal problem is studied in [18,19]. However, McLaughlin's uniqueness theorem does not hold for the vectorial Sturm-Liouville problems.…”

“…However, McLaughlin's uniqueness theorem does not hold for the vectorial Sturm-Liouville problems. To clarify our problem, the following definitions could be seen in [18,19]. For the convenience of reader, we state here again.…”

“…It seems to be the first study of the vectorial inverse nodal problem. Cheng et al generalized to the arbitrary separated boundary conditions in [19]. Chan [6] investigated some eigenvalue problems of (1).…”

Recovering the Sturm-Liouville Operator with Singular Potential Using Nodal Data

“…Inverse problems of such a type are called inverse nodal problems. For the classical SturmLiouville operator with q = σ ∈ L 1 (0, π) such a problem was investigated in detail [1][2][3][4].…”

Reconstruction of Potential Function for Diffusion Operator

“…The known results contain the uniqueness [2,10,14], the reconstruction formula [3,4,6,16], and the numerical results [8]. As a generalization of the Sturm-Liouville equation, vectorial Sturm-Liouville equations were also solved by using the nodal points of eigenfunctions [5].…”