Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 13047

Special Issue Editor


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Guest Editor
1. Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, 443086 Samara, Russia
2. Senior Researcher, Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, 410012 Saratov, Russia
3. S.M. Nikolskii Mathematical Institute, Peoples' Friendship University of Russia (RUDN University), Miklukho-Maklaya Street 6, 117198 Moscow, Russia
Interests: inverse spectral problems; ordinary differential equations; functional analysis; Sturm-Liouville problems; differential operators on graphs; differential operators with distribution coefficients; partial inverse problems

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to the spectral theory of ordinary differential and functional–differential operators. Both direct and inverse spectral problems are included. Such problems play a fundamental role in mathematics and have applications in various fields of science and engineering, e.g., in quantum and classical mechanics, geophysics, acoustics, and electronics.

Direct spectral problems consist in studying the properties of spectral characteristics such as asymptotical formulas for eigenvalues and eigenfunctions, trace formulas, completeness and basicity of root functions, eigen convergence theorems, etc.

Inverse spectral problems consist in the recovery of operators from their spectral characteristics. In recent years, the theory of inverse problems has been actively developing not only for differential operators, but also for integro-differential operators, functional–differential operators with delays, with frozen arguments, and other related classes of operators.

The topics of the potential submissions are not limited to the issues mentioned above. Papers on applications of spectral problems (e.g., to linear and nonlinear partial differential equations) are also encouraged. 

Prof. Dr. Natalia P. Bondarenko
Guest Editor

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Keywords

  • direct spectral problems
  • inverse spectral problems
  • scattering problems
  • ordinary differential operators
  • boundary value problems
  • eigenvalue asymptotics
  • basicity of root functions
  • integro-differential operators
  • functional–differential operators with delay
  • functional–differential operators with frozen argument
  • functional–differential operators with involution

Published Papers (17 papers)

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Research

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26 pages, 394 KiB  
Article
Geometric Approximation of Point Interactions in Three-Dimensional Domains
by Denis Ivanovich Borisov
Mathematics 2024, 12(7), 1031; https://doi.org/10.3390/math12071031 - 29 Mar 2024
Viewed by 392
Abstract
In this paper, we study a three-dimensional second-order elliptic operator with a point interaction in an arbitrary domain. The operator is supposed to be self-adjoint. We cut out a small cavity around the center of the interaction and consider an operator in such [...] Read more.
In this paper, we study a three-dimensional second-order elliptic operator with a point interaction in an arbitrary domain. The operator is supposed to be self-adjoint. We cut out a small cavity around the center of the interaction and consider an operator in such perforated domain with the Robin condition on the boundary of the cavity. Our main result states that once the coefficient in this Robin condition is appropriately chosen, the operator in the perforated domain converges to that with the point interaction in the norm resolvent sense. We also succeed in establishing order-sharp estimates for the convergence rate. Full article
21 pages, 330 KiB  
Article
Schatten Index of the Sectorial Operator via the Real Component of Its Inverse
by Maksim V. Kukushkin
Mathematics 2024, 12(4), 540; https://doi.org/10.3390/math12040540 - 08 Feb 2024
Viewed by 434
Abstract
In this paper, we study spectral properties of non-self-adjoint operators with the discrete spectrum. The main challenge is to represent a complete description of belonging to the Schatten class through the properties of the Hermitian real component. The method of estimating the singular [...] Read more.
In this paper, we study spectral properties of non-self-adjoint operators with the discrete spectrum. The main challenge is to represent a complete description of belonging to the Schatten class through the properties of the Hermitian real component. The method of estimating the singular values is elaborated by virtue of the established asymptotic formulas. The latter fundamental result is advantageous since, of many theoretical statements based upon it, one of them is a concept on the root vectors series expansion, which leads to a wide spectrum of applications in the theory of evolution equations. In this regard, the evolution equations of fractional order with the sectorial operator in the term not containing the time variable are involved. The concrete well-known operators are considered and the advantage of the represented method is convexly shown. Full article
13 pages, 269 KiB  
Article
To the Question of the Solvability of the Ionkin Problem for Partial Differential Equations
by Aleksandr I. Kozhanov
Mathematics 2024, 12(3), 487; https://doi.org/10.3390/math12030487 - 02 Feb 2024
Viewed by 451
Abstract
We study the solvability of the Ionkin problem for some differential equations with one space variable. These equations include parabolic and quasiparabolic, hyperbolic and quasihyperbolic, pseudoparabolic and pseudohyperbolic, elliptic and quasielliptic equations and equations of many other types. For the above equations, the [...] Read more.
We study the solvability of the Ionkin problem for some differential equations with one space variable. These equations include parabolic and quasiparabolic, hyperbolic and quasihyperbolic, pseudoparabolic and pseudohyperbolic, elliptic and quasielliptic equations and equations of many other types. For the above equations, the following theorems are proved with the use of the splitting method: the existence of regular solutions—solutions that all have weak derivatives in the sense of S. L. Sobolev and occur in the corresponding equation. Full article
11 pages, 241 KiB  
Article
On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations
by Yaudat T. Sultanaev, Nur F. Valeev and Elvira A. Nazirova
Mathematics 2024, 12(2), 213; https://doi.org/10.3390/math12020213 - 09 Jan 2024
Viewed by 447
Abstract
This work is devoted to the development of methods for constructing asymptotic formulas as x of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression [...] Read more.
This work is devoted to the development of methods for constructing asymptotic formulas as x of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh–Levitan regularity conditions). As a model equation, the fifth-order equation i2p(x)y+p(x)y+q(x)y=λy, along with various behaviors of coefficients p(x),q(x), is investigated. New asymptotic formulas are obtained for the case when the function h(x)=1+p1/2(x)L1[1,) significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied. Full article
17 pages, 346 KiB  
Article
An Inverse Sturm–Liouville-Type Problem with Constant Delay and Non-Zero Initial Function
by Sergey Buterin and Sergey Vasilev
Mathematics 2023, 11(23), 4764; https://doi.org/10.3390/math11234764 - 25 Nov 2023
Cited by 3 | Viewed by 800
Abstract
We suggest a new statement of the inverse spectral problem for Sturm–Liouville-type operators with constant delay. This inverse problem consists of recovering the coefficient (often referred to as potential) of the delayed term in the corresponding equation from the spectra of two boundary [...] Read more.
We suggest a new statement of the inverse spectral problem for Sturm–Liouville-type operators with constant delay. This inverse problem consists of recovering the coefficient (often referred to as potential) of the delayed term in the corresponding equation from the spectra of two boundary value problems with one common boundary condition. The previous studies, however, focus mostly on the case of zero initial function, i.e., they exploit the assumption that the potential vanishes on the corresponding subinterval. In the present paper, we waive that assumption in favor of a continuously matching initial function, which leads to the appearance of an additional term with a frozen argument in the equation. For the resulting new inverse problem, we pay special attention to the situation when one of the spectra is given only partially. Sufficient conditions and necessary conditions on the corresponding subspectrum for the unique determination of the potential are obtained, and a constructive procedure for solving the inverse problem is given. Moreover, we obtain the characterization of the spectra for the zero initial function and the Neumann common boundary condition, which is found to include an additional restriction as compared with the case of the Dirichlet common condition. Full article
31 pages, 6171 KiB  
Article
Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations
by Sergey Kabanikhin, Maxim Shishlenin, Nikita Novikov and Nikita Prokhoshin
Mathematics 2023, 11(21), 4458; https://doi.org/10.3390/math11214458 - 27 Oct 2023
Viewed by 847
Abstract
In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem [...] Read more.
In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations. Full article
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26 pages, 408 KiB  
Article
Spectrum of One-Dimensional Potential Perturbed by a Small Convolution Operator: General Structure
by D. I. Borisov, A. L. Piatnitski and E. A. Zhizhina
Mathematics 2023, 11(19), 4042; https://doi.org/10.3390/math11194042 - 23 Sep 2023
Viewed by 537
Abstract
We consider an operator of multiplication by a complex-valued potential in L2(R), to which we add a convolution operator multiplied by a small parameter. The convolution kernel is supposed to be an element of [...] Read more.
We consider an operator of multiplication by a complex-valued potential in L2(R), to which we add a convolution operator multiplied by a small parameter. The convolution kernel is supposed to be an element of L1(R), while the potential is a Fourier image of some function from the same space. The considered operator is not supposed to be self-adjoint. We find the essential spectrum of such an operator in an explicit form. We show that the entire spectrum is located in a thin neighbourhood of the spectrum of the multiplication operator. Our main result states that in some fixed neighbourhood of a typical part of the spectrum of the non-perturbed operator, there are no eigenvalues and no points of the residual spectrum of the perturbed one. As a consequence, we conclude that the point and residual spectrum can emerge only in vicinities of certain thresholds in the spectrum of the non-perturbed operator. We also provide simple sufficient conditions ensuring that the considered operator has no residual spectrum at all. Full article
7 pages, 252 KiB  
Article
Trace Formulae for Second-Order Differential Pencils with a Frozen Argument
by Yi-Teng Hu and Murat Şat
Mathematics 2023, 11(18), 3996; https://doi.org/10.3390/math11183996 - 20 Sep 2023
Viewed by 538
Abstract
This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, [...] Read more.
This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, the trace formulae have the same form which reveals the impact of the frozen argument. Full article
7 pages, 244 KiB  
Article
Keller–Osserman Phenomena for Kardar–Parisi–Zhang-Type Inequalities
by Andrey B. Muravnik
Mathematics 2023, 11(17), 3787; https://doi.org/10.3390/math11173787 - 04 Sep 2023
Viewed by 545
Abstract
For coercive quasilinear partial differential inequalities containing nonlinearities of the Kardar–Parisi–Zhang type, we find conditions guaranteeing the absence of global positive solutions. These conditions extend both the classical result of Keller and Osserman and its recent Kon’kov–Shishkov generalization. Additionally, they complement the results [...] Read more.
For coercive quasilinear partial differential inequalities containing nonlinearities of the Kardar–Parisi–Zhang type, we find conditions guaranteeing the absence of global positive solutions. These conditions extend both the classical result of Keller and Osserman and its recent Kon’kov–Shishkov generalization. Additionally, they complement the results for the noncoercive case, which had been previously established by the same author. Full article
51 pages, 7044 KiB  
Article
An Approach to Solving Direct and Inverse Scattering Problems for Non-Selfadjoint Schrödinger Operators on a Half-Line
by Vladislav V. Kravchenko and Lady Estefania Murcia-Lozano
Mathematics 2023, 11(16), 3544; https://doi.org/10.3390/math11163544 - 16 Aug 2023
Viewed by 686
Abstract
In this paper, an approach to solving direct and inverse scattering problems on the half-line for a one-dimensional Schrödinger equation with a complex-valued potential that is exponentially decreasing at infinity is developed. It is based on a power series representation of the Jost [...] Read more.
In this paper, an approach to solving direct and inverse scattering problems on the half-line for a one-dimensional Schrödinger equation with a complex-valued potential that is exponentially decreasing at infinity is developed. It is based on a power series representation of the Jost solution in a unit disk of a complex variable related to the spectral parameter by a Möbius transformation. This representation leads to an efficient method of solving the corresponding direct scattering problem for a given potential, while the solution to the inverse problem is reduced to the computation of the first coefficient of the power series from a system of linear algebraic equations. The approach to solving these direct and inverse scattering problems is illustrated by several explicit examples and numerical testing. Full article
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16 pages, 1781 KiB  
Article
Direct Method for Identification of Two Coefficients of Acoustic Equation
by Nikita Novikov and Maxim Shishlenin
Mathematics 2023, 11(13), 3029; https://doi.org/10.3390/math11133029 - 07 Jul 2023
Cited by 1 | Viewed by 690
Abstract
We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein [...] Read more.
We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein approach, which allows us to obtain both functions by solving two sets of integral equations. The main advantage of the proposed approach is that the method does not use the multiple solution of direct problems, and thus has quite low CPU time requirements. We also consider the variation of the method for the 1D case, where the variation of the wave equation is considered. We illustrate the results with numerical experiments in the 1D and 2D case and study the efficiency and stability of the approach. Full article
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15 pages, 284 KiB  
Article
Higher Monotonicity Properties for Zeros of Certain Sturm-Liouville Functions
by Tzong-Mo Tsai
Mathematics 2023, 11(12), 2787; https://doi.org/10.3390/math11122787 - 20 Jun 2023
Viewed by 667
Abstract
In this paper, we consider the differential equation y+ω2ρ(x)y=0, where ω is a positive parameter. The principal concern here is to find conditions on the function [...] Read more.
In this paper, we consider the differential equation y+ω2ρ(x)y=0, where ω is a positive parameter. The principal concern here is to find conditions on the function ρ1/2(x) which ensure that the consecutive differences of sequences constructed from the zeros of a nontrivial solution of the equation are regular in sign for sufficiently large ω. In particular, if cνk(α) denotes the kth positive zero of the general Bessel (cylinder) function Cν(x;α)=Jν(x)cosαYν(x)sinα of order ν and if |ν|<1/2, we prove that (1)mΔm+2cνk(α)>0(m=0,1,2,;k=1,2,), where Δak=ak+1ak. This type of inequalities was conjectured by Lorch and Szego in 1963. In addition, we show that the differences of the zeros of various orthogonal polynomials with higher degrees possess sign regularity. Full article
17 pages, 350 KiB  
Article
A Class of Quasilinear Equations with Distributed Gerasimov–Caputo Derivatives
by Vladimir E. Fedorov and Nikolay V. Filin
Mathematics 2023, 11(11), 2472; https://doi.org/10.3390/math11112472 - 27 May 2023
Viewed by 696
Abstract
Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the [...] Read more.
Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the Lipschitz continuity condition in phase variables and two types of continuity over all variables of a nonlinear operator in the equation, we obtain two versions on a theorem on the nonlocal existence of a unique solution. Two similar versions of local unique solvability of the Cauchy problem are proved under the local Lipschitz continuity condition for the nonlinear operator. The general results are used for the study of an initial boundary value problem for a generalization of the nonlinear phase field system of equations with distributed derivatives with respect to time. Full article
10 pages, 422 KiB  
Article
Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem
by Yong Tang, Haoze Ni, Fei Song and Yuping Wang
Mathematics 2022, 10(22), 4204; https://doi.org/10.3390/math10224204 - 10 Nov 2022
Viewed by 1015
Abstract
In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of the potential function [...] Read more.
In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of the potential function for the boundary value problem by the first kind Chebyshev wavelet method. Two numerical examples show that the first kind Chebyshev wavelet method for solving the inverse nodal problems for the boundary value problem is valid. Full article
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21 pages, 363 KiB  
Article
The Partial Inverse Spectral and Nodal Problems for Sturm–Liouville Operators on a Star-Shaped Graph
by Xian-Biao Wei, Yan-Hsiou Cheng and Yu-Ping Wang
Mathematics 2022, 10(21), 3971; https://doi.org/10.3390/math10213971 - 26 Oct 2022
Cited by 1 | Viewed by 781
Abstract
We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from [...] Read more.
We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from paired-dense nodal subsets in interior subintervals having a central vertex. In particular, we obtain some uniqueness theorems by replacing the information of nodal data on some fixed edge with part of the eigenvalues under some conditions. Full article
32 pages, 493 KiB  
Article
Reconstruction of Higher-Order Differential Operators by Their Spectral Data
by Natalia P. Bondarenko
Mathematics 2022, 10(20), 3882; https://doi.org/10.3390/math10203882 - 19 Oct 2022
Cited by 7 | Viewed by 984
Abstract
This paper is concerned with inverse spectral problems for higher-order (n>2) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our [...] Read more.
This paper is concerned with inverse spectral problems for higher-order (n>2) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our approach is based on the reduction of an inverse problem to a linear equation in the Banach space of bounded infinite sequences. This equation is derived in a general form that can be applied to various classes of differential operators. The unique solvability of the linear main equation is also proved. By using the solution of the main equation, we derive reconstruction formulas for the differential expression coefficients in the form of series and prove the convergence of these series for several classes of operators. The results of this paper can be used for the constructive solution of inverse spectral problems and for the investigation of their solvability and stability. Full article
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Review

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44 pages, 630 KiB  
Review
Partial Inverse Sturm-Liouville Problems
by Natalia P. Bondarenko
Mathematics 2023, 11(10), 2408; https://doi.org/10.3390/math11102408 - 22 May 2023
Cited by 2 | Viewed by 981
Abstract
This paper presents a review of both classical and modern results pertaining to partial inverse spectral problems for differential operators. Such problems consist in the recovery of differential expression coefficients in some part of the domain (a finite interval or a geometric graph) [...] Read more.
This paper presents a review of both classical and modern results pertaining to partial inverse spectral problems for differential operators. Such problems consist in the recovery of differential expression coefficients in some part of the domain (a finite interval or a geometric graph) from spectral characteristics, while the coefficients in the remaining part of the domain are known a priori. Usually, partial inverse problems require less spectral data than complete inverse problems. In this review, we pay considerable attention to partial inverse problems on graphs and to the unified approach based on the reduction of partial inverse problems to Sturm-Liouville problems with entire analytic functions in a boundary condition. We not only describe the results of selected studies but also compare them with each other and establish interconnections. Full article
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