Spectral Properties of Ordinary Differential Operators with Involution

Vladykina, V. E.; Шкаликов, А. А.

doi:10.1134/s1064562419010046

Cited by 16 publications

(11 citation statements)

References 11 publications

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“…Proof. The isospectrality of the operators L 0 and L 3,p is determined by substituting the general solution (12) of differential equation 9into multipoint conditions (29), (31).…”

Section: Remarkmentioning

confidence: 99%

“…Introduction and main results. During recent years the number of publications with the use of an involution operator in the theory of P T -symmetric operators, P T -symmetric quantum theory (see [11], [12]), various sections of the theory of ordinary differential equations (see [1], [4], [15], [17], [19], [21]- [24], [27], [31], [32]), partial differential equations (see [6], [10], [14], [20], [25]) and differential equations with operator coefficients (see [2], [3], [7], [8]) increased significantly.…”

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confidence: 99%

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The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

Baranetskij¹,

Kalenyuk²,

Kopach³

et al. 2020

Mat. Stud.

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The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

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“…Proof. The isospectrality of the operators L 0 and L 3,p is determined by substituting the general solution (12) of differential equation 9into multipoint conditions (29), (31).…”

Section: Remarkmentioning

confidence: 99%

mentioning

confidence: 99%

The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

Baranetskij¹,

Kalenyuk²,

Kopach³

et al. 2020

Mat. Stud.

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“…In recent years, the spectral theory of ordinary FDO with involution has been actively developed. Vladykina and Shkalikov [6,7] have described the classes of regular boundary conditions and studied spectral properties for the arbitrary-order FDO of form y (n) (−x) + n−1 j=0 p j (x)y (j) (−x) + αy (m) (x) + m−1 j=0 q j (x)y (j) (x),…”

Section: Introductionmentioning

confidence: 99%

Inverse spectral problems for functional-differential operators with involution

Bondarenko¹

2021

Preprint

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The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems.

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“…Sadybekov and A.M. Sarsenby [36][37][38][39][40][41]. Over the past decade, researchers' interest in differential equations with deviating arguments has grown markedly, as evidenced by publications [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. Theory of bases is described in detail in [60][61][62][63].…”

Section: Introductionmentioning

confidence: 99%