Eigenvalues of fourth-order boundary value problems with distributional potentials

Zhang, Haiyan; Ao, Ji-jun; Bo, Fang-zhen

doi:10.3934/math.2022407

Cited by 8 publications

(4 citation statements)

References 30 publications

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“…Lemma 5.8 is proved by the well-known method for obtaining continuous dependence of the spectral data on the boundary value problem coefficients (see [35]). In recent years, this method has been actively developed for various classes of differential operators (see, e.g., [6,36]). Therefore, here we outline the proof of Lemma 5.8 briefly.…”

Section: Proofs Of Theorems 23 and 24mentioning

confidence: 99%

Inverse spectral problem for the third-order differential equation

Bondarenko¹

2023

Preprint

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This paper is concerned with the inverse spectral problem for the third-order differential equation with distribution coefficient. The inverse problem consists in the recovery of the differential expression coefficients from the spectral data of two boundary value problems with separated boundary conditions. For this inverse problem, we solve the most fundamental question of the inverse spectral theory about the necessary and sufficient conditions of solvability. In addition, we prove the local solvability and stability of the inverse problem. Furthermore, we obtain very simple sufficient conditions of solvability in the self-adjoint case. The main results are proved by a constructive method that reduces the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. In the future, our results can be generalized to various classes of higher-order differential operators with integrable or distribution coefficients.

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Section: Proofs Of Theorems 23 and 24mentioning

confidence: 99%

Inverse spectral problem for the third-order differential equation

Bondarenko¹

2023

Preprint

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“…The Schrödinger operators with distribution potentials are widely used in quantum mechanics for describing the interaction between individual particles 9 . Some aspects of spectral theory for the fourth‐order differential operators with distribution coefficients were recently investigated in Ugurlu and Bairamov 10 and Zhang et al 11 The development of the general theory for higher order differential operators with distribution coefficients could unify the approaches to specific problems in various applications, as well as causes interest from the purely mathematical point of view.…”

Section: Introductionmentioning

confidence: 99%

Linear differential operators with distribution coefficients of various singularity orders

Bondarenko

2022

Math Methods in App Sciences

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In this paper, the linear differential expression of order n ≥ 2 with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we present new statements of inverse spectral problems that consist in the recovery of differential operators from the Weyl matrix on the half-line and on a finite interval. The uniqueness theorems for these two inverse problems are proved by developing the method of spectral mappings. To the best of the author's knowledge, inverse problems for higher order differential operators with distribution coefficients on the half-line have not been studied before. For the finite interval case, we for the first time consider the issue of recovering coefficients of the boundary conditions.

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“…The Schrödinger operators with distribution potentials are widely used in quantum mechanics for describing the interaction between individual particles [8]. Some aspects of spectral theory for the fourth-order differential operators with distribution coefficients were recently investigated in [9,10]. The development of the general theory for higher-order differential operators with distribution coefficients could unify the approaches to specific problems in various applications, as well as causes interest from the purely mathematical point of view.…”

Section: Introductionmentioning

confidence: 99%