Inverse nodal problems on quantum tree graphs

In this study, some properties of the pencils of singular Sturm–Liouville operators are investigated. Firstly, the behaviors of eigenvalues and eigenfunctions is learned, then for each discontinuity point a solution of the inverse problem is given to determine the potential function and parameters , and with the help of a dense set of nodes. And finally, a constructive method is given for solving the given inverse problem.

“…In Koyunbakan and Mosazadeh [34], a Sturm–Liouville operator with boundary conditions dependent on the spectral parameter and discontinuous conditions at

a&amp;amp;#x0003D;\frac{\pi }{2}

Section: Introductionmentioning

confidence: 99%

Inverse nodal problems for singular diffusion equation

Amirov,

Durak

2024

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Oviedo

Pinasco

2023

In this work, we study the nodal inverse problem for Sturm–Liouville operators on quantum star graphs. We show that the zeros of eigenfunctions characterize the potential, and we present a simple algorithm that enables us to approximate it. The method does not rely on a priori information about the asymptotic behavior of eigenvalues. We manage to reduce the problem to a family of constant coefficient eigenvalue problems that can be solved explicitly and contain enough information to recover the potential. We also consider the multiplicity of eigenvalues and their vanishing properties, which only hold generically.

“…Compared to the direct problem, the inverse SLP is both attractive and open to improvement [24, 25]. In the inverse problem, parameters such as spectrum, norming constants, zeros of eigenfunctions, and Weyl function are given, and the coefficient of the equation

q(x)

, the potential

q

, and the coefficients in the boundary conditions are found [26–29]. Normally, two spectra or a spectrum and norming constants are sufficient to find the potential

q

as uniquely.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of the potential in conformable Sturm‐Liouville problem

Koyunbakan

2023

This study is about the inverse problem of the conformable Sturm‐Liouville problem SLP. Namely, we first defined the problem with separable boundary conditions and gave some of its spectral properties. Then, by spectral data, we showed that the potential and the constants in the boundary conditions are coincident for two different problems. Here, the potential is in the space . We have also shown that if the potential function has the property of being symmetrical, only one spectrum is sufficient to find this function as uniquely. It is worth remembering that the methods and calculations used here are similar to those in the SLP with classical derivative. However, we think results will be valuable in Sturm‐Liouville theory.