S. Albeverio, R. Hryniv, Ya. Mykytyuk
The spectral properties of Dirac operators on $(0,1)$ with potentials that belong entrywise to $L_p(0,1)$, for some $p\in[1,\infty)$, are studied. The algorithm of reconstruction of the potential from two spectra or from one spectrum and the corresponding norming constants is established, and a complete solution of the inverse spectral problem is provided.
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We consider the direct and inverse spectral problems for Dirac operators on $(0,1)$ with matrix-valued potentials whose entries belong to $L_p(0,1)$, $p\in[1,\infty)$. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest a method for reconstructing the potential from the corresponding spectral data.
Inverse problem for Dirac systems with locally square summable potentials and rectangular Weyl functions is solved. For that purpose we use a new result on the linear similarity between operators from a subclass of triangular integral operators and the operator of integration.
We describe the necessary and sufficient conditions for two sequences {μ_n}^\infty_n=0 and {a_n}^\infty_n=0 to be correspondingly the set of eigenvalues and the set of norming constants of a Sturm-Liouville problem with real summable potential q and in advance fixed separated boundary conditions.
The paper deals with the Dirac operator generated on the finite interval $[0,π]$ by the differential expression $-B\mathbf{y}'+Q(x)\mathbf{y}$, where $$ B=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad Q(x)=\begin{pmatrix}q_1(x)&q_2(x)\\q_3(x)&q_4(x)\end{pmatrix}, $$ and the entries $q_j(x)$ belong to~$L_p[0,π]$ for some $p\geqslant 1$. The classes of regular and strongly regular operators of this form are defined, depending on the boundary conditions. The asymptotic formulas for the eigenvalues and eigenfunctions of such operators are obtained with remainders depending on~$p$. It it is proved t…
We deal with the Dirac operator $\mathcal L_{P,U}$ generated in the space $\mathbb H=(L_2[0,π])^2$ by differential expression \begin{gather*} \ell_P(\mathbf y)=B\mathbf y'+P\mathbf y,\quad B = \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}, \qquad P(x) = \begin{pmatrix} p_1(x) & p_2(x) \\ p_3(x) & p_4(x) \end{pmatrix}, \qquad \mathbf y(x)=\begin{pmatrix}y_1(x)\\ y_2(x)\end{pmatrix}, \end{gather*} and regular boundary conditions $$ U(\mathbf y)=\begin{pmatrix}u_{11} & u_{12}\\ u_{21} & u_{22}\end{pmatrix}\begin{pmatrix}y_1(0)\\ y_2(0)\end{pmatrix}+\begin{pmatrix}u_{13} & u_{14}\\ u_{23} & u_{24}…
On a necessary condition of convergence of spectral expansions corresponding to the Dirac operators
本文针对复值可和势的Dirac算子，建立了其根函数系谱展开收敛的一个必要条件。
On the completeness of root function system of the Dirac operator with two-point boundary conditions
本文研究带可和复值位势和非正则两点边界条件的Dirac算子，证明了其根函数系的完备性，并给出基本解系的显式表达式。
Spectral problem for the Dirac operator with regular but not strongly regular boundary conditions and complex-valued potential summable over a finite interval is considered. The purpose of this paper is to find conditions under which the root function system forms a usual Riesz basis rather than a Riesz basis with parentheses.
On the spectrum of $2\times 2$ Dirac operator with degenerate boundary conditions
本文研究了具有退化边界条件和复值可和势的 $2\ imes 2$ Dirac 算子的谱问题，并找到了使该问题谱与无微扰算子谱一致的充分条件。
We consider the self-adjoint Dirac operators on a finite interval with summable matrix-valued potentials and general boundary conditions. For such operators, we study the inverse problem of reconstructing the potential and the boundary conditions of the operator from its eigenvalues and suitably defined norming matrices.
On the completeness of root function system of the $2\times 2$ Dirac operators with non-regular boundary conditions
本文研究了具有可和复值势及非正则边界条件的 $2\ imes 2$ Dirac 算子根函数系的完备性，并建立了该算子根函数系完备的充分条件。
New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg-Marchenko type uniqueness theorems are derived for both discrete and continuous non-self-adjoint systems too.
We consider the Sturm-Liouville operator Lu=u''-q(x)u with regular but not strongly regular boundary conditions. Under some supplementary assumptions we prove that the set of potentials q(x) that ensure an asymptotically multiple spectrum is everywhere dense in the space of summable functions.
In this paper inverse problems for Dirac operator with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are provided, which are generalizations of the well-known Weyl function and Borg's inverse problem for the classical Dirac operator.
We consider massless Dirac operators on the half-line with compactly supported potentials. We solve the inverse problems in terms of Jost function and scattering matrix (including characterization). We study resonances as zeros of Jost function and prove that a potential is uniquely determined by its resonances. Moreover, we prove the following: 1) resonances are free parameters and a potential continuously depends on a resonance, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonance counting function is determined, 4) these results are applied to canonical systems.
In this paper, we consider a problem for the first order Dirac differential equations system with spectral parameter dependent in boundary condition. The asymptotic behaviors of eigenvalues, eigenfunctions and normalizing numbers of this system are investigated. The expansion formula with respect to eigenfunctions is obtained and Parseval equality is given. The main theorem on necessary and sufficient conditions for the solvabilty of inverse problem is proved and the algorithm of reconstruction of potential from spectral data (the sets of eigenvalues and normalizing numbers) is given.
New inverse and half-inverse problems: {\it sliding problems} are introduced. In this way several physically important equations are recovered from the quantum defect. In particular, sliding problems are solved for radial Schrödinger equation, radial Dirac system and multidimensional Schrödinger equation. Systems with Coulomb-type potentials are considered as well.
We prove that the inverse spectral mapping reconstructing the square integrable potentials on [0,1] of Dirac operators in the AKNS form from their spectral data (two spectra or one spectrum and the corresponding norming constants) is analytic and uniformly stable in a certain sense.
Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl-Titchmarsh-Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one regular endpoint. Moreover, our result also improves the currently known results for canonical (Hamiltonian) systems. If one endpoint is limit circle case, we also establish corresponding two-spectra re…
We prove the Cesàro boundedness of eigenfunctions of the Dirac operator on the half-line with a square-summable potential. The proof is based on the theory of Krein systems and, in particular, on the continuous version of a theorem by A. Mate, P. Nevai and V. Totik from 1991.
The paper is concerned with the Riesz basis property of a boundary value problem associated in $L^2[0,1] \otimes \mathbb{C}^2$ with the following $2 \times 2$ Dirac type equation $$ L y = -i B^{-1} y' + Q(x) y = λy, \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \quad (1) $$ with a summable potential matrix $Q \in L^1[0,1] \otimes \mathbb{C}^{2 \times 2}$ and $b_1 < 0 < b_2$. If $b_2 = -b_1 =1$ this equation is equivalent to one dimensional Dirac equation. It is proved that the system of root functions of a linear boundary value …
A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg-Marchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schrödinger equation are also derived.
We establish simple connections between response functions of the dynamical Dirac systems and $A$-amplitudes and Weyl functions of the spectral Dirac systems. Using these connections we propose a new and rigorous procedure to recover a general-type dynamical Dirac system from its response function as well as a procedure to construct explicit solutions of this problem.
Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions
研究通过谱函数恢复区间或半轴上Dirac系统的反问题，提出局部平方可积势的必要充分条件，并探讨标量谱函数与Paley-Wiener采样测度的联系。
Resonances and inverse problems for energy-dependent potentials on the half-line
本文研究了半直线上紧支撑且线性能量依赖势的薛定谔方程，给出了特征值与共振数目的估计，解决了无特征值情况下的逆共振问题，并利用薛定谔与Dirac方程的对应关系简化了散射问题。
We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of $\mathcal G$. Special attention is paid to Robin matching conditions with parameter $τ\in\mathbb R\cup\{\infty\}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference…
We consider inverse dynamic and spectral problems for the one dimensional Dirac system on a finite tree. Our aim will be to recover the topology of a tree (lengths and connectivity of edges) as well as matrix potentials on each edge. As inverse data we use the Weyl-Titchmarsh matrix function or the dynamic response operator.
The inverse nodal problem for Dirac differential operator perturbated by a Volterra integral operator is studied. We prove that dense subset of the nodal points determines the coefficients of differential and integral part of the operator. We also provide a uniqueness theorem and an algorithm to reconstruct the coefficients of the problem by using the nodal points.
The principal objective in this paper is a new inverse approach to general Dirac-type systems modeled after B. Simon's 1999 inverse approach to half-line Schrödinger operators. In particular, we derive the so-called A-equation associated to Dirac-type systems and, given a fundamental positivity condition, we prove that this integro-differential equation for A is uniquely solvable. We show how to recover the matrix-valued potential coefficient from A. This approach is also applicable to matrix-valued Schrödinger operators on a half-line.
Inverse problem for Dirac operators with a small delay
本文研究时滞小于区间长度三分之一的Dirac型算子逆谱问题，证明两个谱不足以完全恢复势函数，并讨论了Ambarzumian型逆问题。
In this work, the Dirac-type integro di§erential system with one classical boundary condition and another nonlocal integral boundary condition is considered. We obtain the asymptotic formulae for the solutions, eigenvalues and nodal points. We also investigate the inverse nodal problem and prove that given a dense subset of nodal points uniquely determines the boundary condition parameter and the potential function of the considered di§erential system. We also provide an e§ective procedure for solving the inverse nodal problem.
This paper is related to an inverse problem for a class of Dirac operators with discontinuous coefficient and eigenvalue parameter contained in boundary conditions. The asymptotic formula of eigenvalues of this problem is examined. The theorem on completeness of eigenfunctions is proved. The expansion formula with respect to eigenfunctions is obtained and Parseval equality is given. Weyl solution and Weyl function are constructed. Uniqueness theorem of the inverse problem respect to the Weyl function is proved.
Self-adjoint Dirac systems and subclasses of canonical systems, which generalize Dirac systems are studied. Explicit and global solutions of direct and inverse problems are obtained. A local Borg-Marchenko-type theorem, integral representation of the Weyl function, and results on interpolation of Weyl functions are also derived.
An integro-differential Dirac system with an integral term in the form of convolution is considered. We suppose that the convolution kernel is known a priori on a part of the interval, and recover it on the remaining part, using a part of the spectrum. We prove the uniqueness theorem, provide an algorithm for the solution of the inverse problem together with necessary and sufficient conditions for its solvability.
The paper is concerned with the following $n\times n$ Dirac type equation$$Ly=-iB(x)^{-1}(y'+Q(x)y)=λy, \quad B(x)=B(x)^*,\quad y={\rm col}(y_1,\ldots,y_n),\quad x\in[0,\ell],$$ on a finite interval $[0,\ell]$. Here $Q$ is a summable potential $n\times n$ matrix function and $B$ is an invertible self-adjoint diagonal bounded matrix function. If $n=2m$ and $B(x)={\rm diag}(-I_m,I_m)$, this equation is equivalent to Dirac equation of order $n$. We show the existence of triangular transformation operators for such equation under additional uniform separation conditions on the entries of the matri…
Solving Inverse Dirac-weighted Sturm-Liouville Problems via Cauchy problems
本文基于点相互作用方法，通过构造柯西问题显式重构带单点、两点及多点Dirac权的正则Sturm-Liouville问题的势函数，并讨论多点情形。
Incomplete inverse problem for Dirac operator with constant delay
本文研究了具有特定常数时滞的 Dirac 型算子的不完全逆谱问题，在已知子区间势函数的前提下，利用两个边值问题的完整谱重构整个区间的势函数，并证明了解的唯一性与构造性方法。
Inverse nodal problems for Dirac differential operators with jump condition
本文研究了带跳跃条件的 Dirac 微分算子的逆结点问题，提出求解渐近表达式的新方法，证明系数可由结点稠密子集唯一确定，并给出构造解的算法。
Discrete Dirac type self-adjoint system is equivalent to the block Szegö recurrence. Representation of the fundamental solution is obtained, inverse problems on the interval and semi-axis are solved. A Borg-Marchenko type result is obtained too. Connections with the block Toeplitz matrices are treated.
Inverse problems for Dirac operators with a constant delay less than half of the interval
本文研究常数时滞在区间五分之二到一半之间的Dirac型算子的逆谱问题，通过两类边值问题的完整谱重构两个复L2-势，并给出唯一性、可解性条件、重构算法及一致稳定性。
Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form $ϕ(λ)\exp\{-2iλD\}$, where $ϕ$ is a strictly proper rational matrix function and $D=D^* \geq 0$ is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integra…
Direct and inverse spectral continuity for Dirac operators
本文研究了半直线 Dirac 算子谱数据的同胚连续性，通过基于 Schur 算法求解半格上具 $δ$-相互作用算子的逆谱问题，首次证明了一般 $L^2$-情形下的显式双边一致估计。
A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szegö recurrence, Schur coefficients and structured matrices are treated. Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the interval and semiaxis are solved.
In this work, a complete solution of the inverse spectral problem for a class of Dirac differential equations system is given by spectral data (eigenvalues and normalizing numbers). As a direct problem, the eigenvalue problem is solved: the asymptotic formulas of eigenvalues, eigenfunctions and normalizing numbers of problem are obtained, spectral data is defined by the sets of eigenvalues and normalizing numbers. The expansion formula with respect to eigenfunctions is obtained. Gelfand-Levitan-Marchenko equation is derived. The main theorem on necessary and sufficient conditions for the solva…
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…
We study the non-selfadjoint Dirac system on a finite interval having non-integrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and the inverse spectral problem is investigated. We provide a constructive procedure for the solution of the inverse problem, and prove its uniqueness. Moreover, necessary and sufficient conditions for the global solvability of this nonlinear inverse problem are obtained.
Inverse nodal problem on Dirac operator is finding the parameters in the boundary conditions, the number m and the potential function V in the Dirac equations by using a set of nodal points of a component of two component vector eigenfunctions as the given spectral data. In this study, we solve a stability problem using nodal set of vector eigenfunctions and show that space of all V functions is homeomorphic to the partition set of all space of asymptotically equivalent nodal sequences induced by an equivalence relation. Furthermore, we give a reconstruction formula for the potential function …
Weyl theory and explicit solutions of direct and inverse problems for a Dirac system with rectangular matrix potential
针对矩形矩阵势的Dirac系统，发展了非经典Weyl理论，引入Weyl函数并处理正问题，同时获得了有理Weyl矩阵函数情形下正反问题的显式解。
We consider the direct and inverse spectral problems for Dirac operators that are generated by the differential expressions $$ \mathfrak t_q:=\frac{1}{i}[I&0 0&-I]\frac{d}{dx}+[0&q q^*&0] $$ and some separated boundary conditions. Here $q$ is an $r\times r$ matrix-valued function with entries belonging to $L_2((0,1),\mathbb C)$ and $I$ is the identity $r\times r$ matrix. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest an algorithm of reconstructing the potential $q$ from the correspondi…
A procedure to recover explicitly self-adjoint matrix Dirac systems on semi-axis (with both discrete and continuous components of spectrum) from rational Weyl functions is considered. Its stability is proved. GBDT version of Baecklund-Darboux transformation and various important results on Riccati equations are used for this purpose.
Procedures to recover explicitly discrete and continuous skew-selfadjoint Dirac systems on semi-axis from rational Weyl matrix functions are considered. Their stability is shown. Some new facts on asymptotics of pseudo-exponential potentials (i.e., of explicit solutions of inverse problems) are proved as well. GBDT version of Backlund-Darboux transformation, methods from system theory and results on algebraic Riccati equations are used for this purpose.
We consider discrete self-adjoint Dirac systems determined by the potentials (sequences) $\{C_k\}$ such that the matrices $C_k$ are positive definite and $j$-unitary, where $j$ is a diagonal $m\times m$ matrix and has $m_1$ entries $1$ and $m_2$ entries $-1$ ($m_1+m_2=m$) on the main diagonal. We construct systems with rational Weyl functions and explicitly solve inverse problem to recover systems from the contractive rational Weyl functions. Moreover, we study the stability of this procedure. The matrices $C_k$ (in the potentials) are so called Halmos extensions of the Verblunsky-type coeffic…
We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.
Spectral Theory of Dirac Operators
该书系统梳理Dirac算子谱理论的核心议题，涵盖变换算子、谱渐近、自伴性、特征函数展开及反问题的唯一性与构造性求解等内容。
In this paper we study direct and inverse problems for discrete and continuous time skew-selfadjoint Dirac systems with rectangular (possibly non-square) pseudo-exponential potentials. For such a system the Weyl function is a strictly proper rational rectangular matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a no…
The inverse nodal problem for Dirac type integro-differential operator with the spectral parameter in the boundary conditions is studied. We prove that dense subset of the nodal points determines the coefficients of differential part of operator and gives partial information for integral part of it.
Weighted eigenvalues of Dirac operators: complete continuity and comparison
本文研究了Dirac算子的加权特征值问题，通过极小极大刻画证明了特征值与特征空间关于逆权重弱$L^p$收敛的连续性，并在无调和旋量条件下建立了比较结果。
We study the non-selfadjoint Dirac system on the line having an non-integrable regular singularity in an interior point with additional matching conditions at the singular point. Special fundamental systems of solutions are constructed with prescribed analytic and asymptotic properties. Behavior of the corresponding Stockes multipliers is established. These fundamental systems of solutions will be used for studying direct and inverse problems of spectral analysis.
We consider 1d-Dirac operator $\mathcal L_{P,U}$ acting in $\mathbb H=(L_2[0,π])^2$ \begin{gather*} \ell(\mathbf y) = B\mathbf y + P(x)\mathbf y,\qquad B = \begin{pmatrix}-i&0\\0&i\end{pmatrix},\\ P(x) = \begin{pmatrix}p_1(x)&p_2(x)\\ p_3(x)&p_4(x) \end{pmatrix},\qquad\mathbf y = \begin{pmatrix}y_1(x)\\ y_2(x)\end{pmatrix} \end{gather*} with arbitrary regular boundary conditions $U(\mathbf y)=0$. The functions $p_j(x)$, $1\le j\le 4$, assumed to be complex valued and summable. Any regular 1d-Dirac operator of such kind has purely discrete spectrum $\{λ_n\}_{n\in\mathbb Z}$, $λ_n=n+O(1)$ as $|n…
On recovering Dirac operators with two delays
针对含两个常数延迟的 Dirac 型泛函微分算子，本文研究其逆谱问题，并给出由四个谱唯一恢复算子的延迟条件及不可能性边界。
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