Inverse nodal problem for polynomial pencil of Sturm‐Liouville operator

Abstract

The paper is about boundary value problem for polynomial pencil of Sturm‐Liouville operators. Especially, we find all coefficients of the operator by using nodal points (zeros of eigenfunctions). Regularly, we find eigenvalues, nodal points, and nodal lengths by Prüfer substitution. These results are used to give a reconstruction formula for all complex functions qd(x), d=true0,n−1‾, which are known potentials in the theory. However, method is similar with some papers; our results more general then because of including many potential functions.

Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

“…Note that the choice of the square root branch for Θ(x) and Λ(x) is uniquely specified by the continuity of these functions, the condition Θ(0) = 1, and (29). If Θ(x) = 0 for some x ∈ [0, π], one can apply the step-by-step process described in [20].…”

“…It is also worth mentioning that, in recent years, a number of scholars have been actively studying inverse problems for quadratic differential pencils (see [28][29][30][31][32][33][34][35][36] and other papers of these authors). The majority of those results are concerned with partial inverse problems, inverse nodal problems, and recovery of the pencils from the interior spectral data.…”

Inverse nodal problems for Sturm–Liouville operators on quantum star graphs

“…Hald [2][3][4]. Several works improved their methods and extended them to other problems and different boundary conditions [5][6][7][8][9][10], the quasilinear p-Laplacian operator [11,12], differential pencils [13,14], eigenvalue depending coefficients or boundary conditions [15,16], and also to quantum graphs [17][18][19][20][21][22][23]. However, most of these works assume the existence of a formula for the asymptotic behavior of eigenvalues or developed it using transmutation operators and Prufer's type transformations.…”