Inverse Spectral Theory Research Articles

An inverse problem for Hankel operators and turbulent solutions of the cubic Szegő equation on the line

We construct inverse spectral theory for finite rank Hankel operators acting on the Hardy space of the upper half-plane. A particular feature of our theory is that we completely characterise the set of spectral data. As an application of this theory, we prove the genericity of turbulent solutions of the cubic Szegő equation on the real line.

Interior inverse problem for global conservative multipeakon solutions of the Camassa-Holm equation

We consider the interior inverse problem associated with the global conservative multipeakon solution of the Camassa-Holm equation. Based on the inverse spectral theory on the half-line and the oscillation property of eigenfunctions, some (non)uniqueness results of the interior inverse problem are obtained. In addition, we give the trace formula, which connects the global conservative multipeakon solution with the corresponding eigenvalues and normalized eigenfunctions.

Inverse problems for Dirac operators with constant delay less than half of the interval

In this work, we consider Dirac-type operators with a constant delay of less than half of the interval and not less than two-fifths of the interval. For our considered Dirac-type operators, two inverse spectral problems are studied. Specifically, reconstruction of two complex L2-potentials is studied from complete spectra of two boundary value problems with one common boundary condition y1(0) = 0 or y2(0) = 0. We give answers to the full range of questions usually raised in the inverse spectral theory. That is, we give uniqueness, necessary and sufficient conditions of the solvability, reconstruction algorithm and uniform stability for our considered inverse problems.

Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties.

Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients

In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order n≥2. We show that every matrix function of this class is associated with some differential expression. Moreover, we construct the family of associated matrices for a fixed differential expression. Furthermore, our regularization results are applied to inverse spectral theory. We study a new type of inverse spectral problems, which consist of the recovery of distribution coefficients from the spectral data independently of the associated matrix. The uniqueness theorems are proved for the inverse problems by the Weyl–Yurko matrix and by the discrete spectral data. As examples, we consider the cases n=2 and n=4 in more detail.

A partial inverse problem for non-self-adjoint Sturm–Liouville operators with a constant delay

Abstract In this paper we study a partial inverse spectral problem for non-self-adjoint Sturm–Liouville operators with a constant delay and show that subspectra of two boundary value problems with one common boundary condition are sufficient to determine the complex potential. We developed the Horváth’s method in [M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc. 353 2001, 10, 4155–4171] for the self-adjoint Sturm–Liouville operator without delay into the non-self-adjoint Sturm–Liouville differential operator with a constant delay.

On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well.

Inverse Problems for Dirac Operators with Constant Delay: Uniqueness, Characterization, Uniform Stability

We initiate studying inverse spectral problems for Dirac-type functional-differential operators with constant delay. For simplicity, we restrict ourselves to the case when the delay parameter is not less than one half of the interval. For the considered case, however, we give answers to the full range of questions usually raised in the inverse spectral theory. Specifically, reconstruction of two complex $$L_{2}$$ -potentials is studied from either complete spectra or subspectra of two boundary value problems with one common boundary condition. We give conditions on the subspectra that are necessary and sufficient for the unique determination of the potentials. Moreover, necessary and sufficient conditions for the solvability of both inverse problems are obtained. For the inverse problem of recovering from the complete spectra, we establish also uniform stability in each ball of a finite radius. For this purpose, we use recent results on uniform stability of sine-type functions with asymptotically separated zeros.

Inverse fractional Sturm‐Liouville problem with eigenparameter in the boundary conditions

Ambarzumyan theorem is one of the first results of the inverse spectral theory. In this work, Ambarzumyan's theorem is proved for a fractional derivative Sturm‐Liouville problem. In addition, a general function that depends on the eigenvalue parameter under boundary conditions is considered.

Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials

In recent years, there appeared a growing interest in the inverse spectral theory for functional‐differential operators with frozen argument. Such operators are nonlocal and belong to the so‐called loaded differential operators, which frequently appear in mathematics as well as natural sciences and engineering. However, to various classes of nonlocal operators, classical methods of the inverse spectral theory are not applicable. For this reason, it is relevant to develop new methods and approaches for solving inverse problems for operators of this type. We establish a deep connection between the inverse problem for operators with frozen argument and Chebyshev polynomials of the first and the second kinds. Appearing to be of an interest from the point of view of the Chebyshev polynomials theory itself, this connection gives a new perspective method for studying inverse problems for operators with frozen argument. In particular, it allows one to completely describe all non‐degenerate and degenerate cases, that is, when the corresponding inverse problem is uniquely solvable or not, respectively. Moreover, it gives a convenient description of all isospectral potentials in the degenerate case, which is demonstrated by some illustrative examples.

Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems

<p style='text-indent:20px;'>This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural.</p>

Recovery of a degenerate space-dependent heat capacity

We are concerned with the reconstruction of the heat capacity coefficient of a one-dimensional heat equation from a sequence of observations of solutions at a single fixed point. This new direct method reconstructs the principal eigenfunction by extracting its Fourier coefficients from the observations of the solution. The algorithm presented here is an alternative to Krein's inverse spectral theory, minimization and iterative procedures. Error estimates and numerical experiments are presented.

Periodic Jacobi operators with complex coefficients

We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any N -th degree polynomial whose leading coefficient is (-1)^N is the Hill discriminant of finitely many discrete N -periodic Schrödinger operators (Theorem 1). Also, in the case where the spectrum is a closed interval we prove a result (Theorem 2) which is the analog of Borg's Theorem for the non-self-adjoint Jacobi case.

An inverse spectral problem for second-order functional-differential pencils with two delays

Recently, there appeared a considerable interest in inverse Sturm–Liouville-type problems with constant delay. However, necessary and sufficient conditions for solvability of such problems were obtained only in one very particular situation. Here we address this gap by obtaining necessary and sufficient conditions in the case of functional-differential pencils possessing a more general form along with a nonlinear dependence on the spectral parameter. For this purpose, we develop the so-called transformation operator approach, which allows reducing the inverse problem to a nonlinear vectorial integral equation. In Appendix A, we obtain as a corollary the analogous result for Sturm–Liouville operators with delay. Remarkably, the present paper is the first work dealing with an inverse problem for functional-differential pencils in any form. Besides generality of the pencils under consideration, an important advantage of studying the inverse problem for them is the possibility of recovering both delayed terms, which is impossible for the Sturm–Liouville operators with two delays. The latter, in turn, is illustrated even for different values of these two delays by a counterexample in Appendix B. We also provide a brief survey on the contemporary state of the inverse spectral theory for operators with delay observing recently answered long-term open questions.

On non-uniqueness of recovering Sturm–Liouville operators with delay

The inverse spectral theory was developed most widely for purely differential operators. Meanwhile, various their nonlocal perturbations are often more adequate for modelling various real-world processes frequently possessing a nonlocal nature. Thus, there recently appeared a growing interest in the nonlinear inverse problems for differential operators with constant delay. However, because of their peculiarity, such problems are still insufficiently studied. In particular, there has been a long-term basic open question about the unique solvability of such inverse problems for small values of the delay. In this paper, we address that question by giving a negative answer in an important and illustrative situation. Our study is based on numerical simulations, which have finally led to a deep theoretical result. Moreover, we develop a general approach for constructing relevant iso-bispectral potentials. The obtained results essentially change the future strategy of studying inverse problems for differential operators with delay.

Schrödinger operators periodic in octants

We consider Schrödinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrödinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.

Eigenvalues of periodic difference operators on lattice octants

Consider a difference operator H with periodic coefficients on the octant of the lattice. We show that for any integer N and any bounded interval I, there exists an operator H having N eigenvalues, counted with multiplicity on this interval, and does not exist other spectra on the interval. Also right and to the left of it are spectra and the corresponding subspaces have an infinite dimension. Moreover, we prove similar results for other domains and any dimension. The proof is based on the inverse spectral theory for periodic Jacobi operators.

On an open question in recovering Sturm–Liouville-type operators with delay

In recent years, there appeared a considerable interest in the inverse spectral theory for functional-differential operators with constant delay. In particular, it is well known that specification of the spectra of two operators ℓj,j=0,1, generated by one and the same functional-differential expression −y′′(x)+q(x)y(x−a) under the boundary conditions y(0)=y(j)(π)=0 uniquely determines the complex-valued square-integrable potential q(x) vanishing on (0,a) as soon as a∈[π∕2,π). For many years, it has been a challenging open question whether this uniqueness result would remain true also when a∈(0,π∕2). Recently, a positive answer was obtained for the case a∈[2π∕5,π∕2). In this paper, we give, however, a negative answer to this question for a∈[π∕3,2π∕5) by constructing an infinite family of iso-bispectral potentials. Some discussion on a possibility of constructing a similar counterexample for other types of boundary conditions is provided, and new open questions are outlined.

An Extension of The First Eigen-type Ambarzumyan theorem

An extension of the first eigenvalue-type Ambarzumyan's theorem are provided for the arbitrary self-adjoint Sturm-Liouville differential operators. The result makes a contribution to the Poschel-Trubowitz inverse spectral theory as well.

Spectral analysis of the matrix Sturm\u2013Liouville operator

The self-adjoint matrix Sturm–Liouville operator on a finite interval with a boundary condition in general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on an analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm–Liouville operators on a star-shaped graph with two different types of matching conditions.