A method for successive synthesis of the Weyl matrix on the square lattice is proposed. It allows one to compute the Weyl matrix of a large graph by adding new edges and solving elementary systems of linear algebraic equations at each step. Synthesis of the Weyl matrix is useful to further study the inverse problems of the square lattice. Moreover, our approach can be extended to other types of periodic lattices.

In this paper, we study a new inverse spectral problem that consists in the recovery of the third-order differential equation from two spectra corresponding to the boundary conditions y(0) = y(1) = y(2) = 0 and y(0) = y'(0) = y(1) = 0. The uniqueness and existence theorems for the solution are obtained. To prove the results, we treat the inverse problem using a general approach that reconstructs higher-order differential operators from the Weyl-Yurko matrix.

In this paper, the Sturm-Liouville problem with nonseparated quasiperiodic boundary conditions is considered. We study the recovery of the problem parameters from the Hill-type discriminant, the Dirichlet spectrum, and the sequence of signs. We obtain the necessary and sufficient conditions of solvability, the local solvability and stability, as well as the uniform stability for this inverse spectral problem.

In this paper, we study differential operators associated with the formal expression $y''' + s(σ' y)' + s σ' y' + κσ'' y$ with distribution coefficient $σ'' \in W_3^{-2}$, where $s$ and $κ$ are constants. The uniqueness theorems are proved for the inverse spectral problems that consist in the recovery of $σ$ from the Weyl-Yurko matrix on a finite interval and on the half-line. In addition, we discuss the reconstruction of $σ$ and formulate some open problems.

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In this paper, the reconstruction of a linear differential operator of arbitrary order $n \ge 2$ is studied by using two types of spectral characteristics: (i) eigenvalues and weight numbers, (ii) $(2n-2)$ spectra. We prove the unconditional uniform stability of these inverse problems, generalizing the results of Savchuk and Shkalikov [Funct. Anal. Appl. 44 (2010), no. 4, 270--285] to $n > 2$. Furthermore, we for the first time obtain sufficient conditions of solvability for the higher-order inverse problem by $(2n-2)$ spectra. By applying our main results, we get new theorems on the necessary and sufficient conditions of solvability and on the uniform stability of the inverse problems for $n = 3$ and $n = 4$. Our approach is based on the method of spectral mappings, which provides a constructive solution of the inverse problems.

ABSTRACT In this paper, we develop a method for synthesis of the Weyl matrix for Schrödinger operators on carbon nano‐structures which are equivalent to the hexagonal lattice. We construct Weyl matrices for graphs that contain any number of hexagons by adding new edges and solving elementary systems of linear algebraic equations at each step. Our method can be applied in numerical simulations for studying inverse spectral problems for differential operators on hexagonal lattices.

In this paper, for the first time, we study the inverse Sturm–Liouville problem with polynomials of the spectral parameter in the first boundary condition and with entire analytic functions in the second one. For the investigation of this new inverse problem, we develop an approach based on the construction of a special vector functional sequence in a suitable Hilbert space. The uniqueness of recovering the potential and the polynomials of the boundary condition from a part of the spectrum is proved. Furthermore, our main results are applied to the Hochstadt–Lieberman-type problems with polynomial dependence on the spectral parameter not only in the boundary conditions but also in discontinuity (transmission) conditions inside the interval. We prove novel uniqueness theorems, which generalize and improve the previous results in this direction. Note that all the spectral problems in this paper are investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. Moreover, our method is constructive and can be developed in the future for numerical solution and for the study of solvability and stability of inverse spectral problems.

This paper deals with the Sturm-Liouville operators with distribution potentials of the space $W_2^{-1}$ on a metric tree. We study an inverse spectral problem that consists in the recovery of the potentials from the characteristic functions related to various boundary conditions. We prove the uniform stability of this inverse problem for potentials in a ball of any fixed radius, as well as the local stability under small perturbations of the spectral data. Our approach is based on a stable algorithm for the unique reconstruction of the potentials relying on the ideas of the method of spectral mappings.

In this paper, the inverse Sturm-Liouville problem with distribution potential and with polynomials of the spectral parameter in one of the boundary conditions is considered. We for the first time prove local solvability and stability of this inverse problem in the general non-self-adjoint case, taking possible splitting of multiple eigenvalues into account. The proof is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of continuous functions on some circular contour. Moreover, we introduce the generalized Cauchy data, which will be useful for investigation of partial inverse Sturm-Liouville problems with polynomials in the boundary conditions. Local solvability and stability of recovering the potential and the polynomials from the generalized Cauchy data are obtained. Thus, the results of this paper include the first existence theorems for solution of the inverse Sturm-Liouville problems with polynomial dependence on the spectral parameter in the boundary conditions in the case of multiple eigenvalues. In addition, our stability results can be used for justification of numerical methods.

In this paper, the uniform stability of the inverse spectral problem is proved for the matrix Sturm-Liouville operator on a finite interval. Namely, we describe the sets of spectral data, on which the inverse spectral mapping is bounded and, consequently, the uniform estimates hold for the differences of the matrix potentials and of the corresponding coefficients of the boundary conditions. Our approach is based on a constructive procedure for solving the inverse problem by developing ideas of the method of spectral mappings. In addition, we apply our technique to obtain the uniform stability of the inverse Sturm-Liouville problem on the star-shaped graph.

In this paper, we develop a new approach to investigation of the uniform stability for inverse spectral problems. We consider the non‐self‐adjoint Sturm–Liouville problem that consists in the recovery of the potential and the parameters of the boundary conditions from the eigenvalues and the generalized weight numbers. The special case of simple eigenvalues, as well as the general case with multiple eigenvalues, is studied. We find various subsets in the space of spectral data, on which the inverse mapping is Lipschitz continuous, and obtain the corresponding unconditional uniform stability estimates. Furthermore, the conditional uniform stability of the inverse problem under a priori restrictions on the potential is studied. In addition, we prove the uniform stability of the inverse problem by the Cauchy data, which are convenient for numerical reconstruction of the potential and for applications to partial inverse problems.

In this paper, we study an inverse spectral operator for the higher-order differential equation (-1)^my^(2m)+ q y = λ y, where q ∈ L^2(0,π). We prove that if q_2 is sufficiently small, the two spectra corresponding to the both Dirichlet boundary conditions and to the Dirichlet-Neumann ones uniquely determine the potential q. The result extends the Borg theorem from the second order to all even higher orders.

In this paper, we consider Barcilon's inverse problem, which consists of the recovery of the fourth-order differential operator from three spectra. We obtain the relationship of Barcilon's three spectra with the Weyl-Yurko matrix. Moreover, we prove the uniqueness theorem for the inverse problem solution by developing the ideas of the method of spectral mappings. Our approach allows us to obtain the result for the general case of complex-valued distributional coefficients. In the future, the methods and the results of this paper can be generalized to differential operators of orders greater than 4 and used for further development of the inverse problem theory for higher-order differential operators.

In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with complex-valued distributional coefficients. For the case of multiple spectra, we first establish the relationship between spectra and the Weyl-Yurko matrix. Secondly, we prove the uniqueness theorem for the solution of the inverse problems. Our approach allows us to obtain results for the general case of complex-valued distributional coefficients.

We consider a class of self-adjoint Sturm-Liouville problems with rational functions of the spectral parameter in the boundary conditions. The uniform stability for direct and inverse spectral problems is proved for the first time for Sturm-Liouville operator pencils with boundary conditions depending on the eigenparameter. Furthermore, we obtain stability estimates for finite data approximations, which are important from the practical viewpoint. Our method is based on Darboux-type transforms and proving of their Lipschitz continuity.

In this paper, we revisit McLaughlin's inverse problem, which consists in the recovery of the fourth-order differential operator from the eigenvalues and two sequences of weight numbers. We for the first time prove the uniqueness for solution of this problem. Moreover, we obtain the interpretation of McLaughlin's problem in the framework of the general inverse problem theory by Yurko for differential operators of arbitrary orders. An advantage of our approach is that it requires neither smoothness of the coefficients nor self-adjointness of the operator. In addition, we establish the connection between McLaughlin's problem and Barcilon's three-spectra inverse problem.

In this paper, the Sturm-Liouville operators on a graph with a cycle are considered. We study the inverse spectral problem, which consists in the recovery of the potentials from several spectra and a sequence of signs related to the quasiperiodic problem on the loop. The local stability of this inverse problem on the whole graph is proved. In addition, we investigate its uniform stability.

We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schr\"odinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.

In this work, we consider the spectral problems for the Sturm–Liouville operators on a caterpillar graph with the standard matching conditions in the internal vertices and the Neumann or the Dirichlet conditions in the boundary vertices. The regularized trace formulae of these operators are established by using the residue techniques of complex analysis.