Finite Spectrum of Sturm–Liouville Problems with Transmission Conditions Dependent on the Spectral Parameter

Matrix Representations for a Class of Eigenparameter Dependent Sturm–Liouville Problems with Discontinuity

“…We study an SLP in which an eigenparameter is contained in both the boundary and interface conditions, regardless of whether it is self-adjoint or non-self-adjoint. We prove that the problem has, at most, M + N + 5 eigenvalues, which is different from the results in [27], where the number of eigenvalues is, at most, M + N + 4. Moreover, we provide an example to illustrate our conclusion (as it turns out, it affects the number of eigenvalues).…”

“…Assume that v i (i = 1, M) is defined in (21). Then, using (26), and utilizing induction on (27), (22) holds. Moreover, ( 23) and ( 24) can be similarly obtained.…”

“…In a recent paper, Ao et al proved that SLPs with interface conditions dependent on the eigenparameter still have a finite spectrum [27]. Here, the following question arises:…”

Inverse Spectral Problem for Sturm-Liouville Operator with Boundary and Jump Conditions Dependent on the Spectral Parameter

“…Recent years, the boundary value problems with eigenparameter-dependent transmission conditions have drown scholars' much attention and have achieved significant progress, including direct and inverse spectral theory and half inverse spectral theory [26]- [35]. In 2005, Akdogan et al investigated the discontinuous Sturm-Liouville problems, where the spectral parameter not only appears in differential equations, but also in boundary conditions and one of the jump conditions, they got the asymptotic approximation of fundamental solutions and the asymptotic formulae for eigenvalues of such problems [27].…”

“…In 2018 and 2021, Bartels et al presented Sturm-Liouville problems with transfer condition Herglotz dependent on the eigenparameter, and showed the Hilbert space formulation of the problem and calculated out the eigenvalue and eigenfunction asymptotic formula on this problem [31] [34]. Zhang et al studied the finite spectrum of Sturm-Liouville problems with both jump conditions dependent on the spectral parameter [35].…”