Incomplete Inverse Spectral and Nodal Problems for Differential Pencils

Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter

“…According to Lemma 3.1 one can choose sufficiently large N such that for all n > N there is exactly the first component y 1 (x, λ n ) of the eigenfunction y(x, λ n ) of the operator L posses precisely n − r 01 − r 11 nodes in the interval (0, π) corresponding to the eigenvalues λ n > 0. Analogous to the proof of [4,Theorem 6] the similar assertion with the same N is also valid for the operator L 2 of the form…”

“…Such problems play an important role in mathematics and have many applications in natural sciences and engineering (see [1,2,15,[20][21][22]29] and the references therein). The inverse nodal problem, first posed and solved by McLaughlin [13,23], is the problem of constructing operators from given nodes (zeros) of their eigenfunctions (refer to [3][4][5]12,14,17,24,[26][27][28]). From the physical point of view this corresponds to finding, e.g., the density of a string or a beam from the zero-amplitude positions of their eigenvibrations.…”

On half inverse problem for differential pencils with the spectral parameter in boundary conditions

“…In the present paper so-called half inverse spectral problem for the pencil L is studied, which consists in recovering the coefficients of (1), (2) from its spectrum {ρ n }, provided that they are known a priori on one half of the segment [0, π]. The uniqueness theorem for this inverse problem is proved, which in the case h 1 = H 1 = 0 was proved in [17]. Here another approach is used, which is based on the interpolation of entire functions and gives a constructive procedure for solving the half inverse problem.…”

Reconstruction of the Volterra-type integro-differential operator from nodal points

“…Yang provided an algorithm to determine the coefficients of the Sturm-Liouville problem by using the given nodal points in [3]. Inverse nodal problems for different types of operators have been extensively well studied in several papers (see [4][5][6][7][8][9][10][11][12][13][14] and [15]). …”