Alexander Makin
The necessary and sufficient conditions are given for a sequence of complex numbers to be the periodic (or antiperiodic) spectrum of non-self-adjoint Dirac operator.
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Under the assumption that $V \in L^2([0,π]; dx)$, we derive necessary and sufficient conditions for (non-self-adjoint) Schrödinger operators $-d^2/dx^2+V$ in $L^2([0,π]; dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues). We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $-d^2/dx^2+V$ in $L^p([0,π]; dx)$, $p \in (1,\infty)$.
We find conditions on the potential of the non-self-adjoint Mathieu-Hill operator such that the all eigenvalues of the periodic, antiperiodic, Dirichlet and Neumann boundary value problems are simple.
The one-dimensional Dirac operator with periodic potential $V=\begin{pmatrix} 0 & \mathcal{P}(x) \\ \mathcal{Q}(x) & 0 \end{pmatrix}$, where $\mathcal{P},\mathcal{Q}\in L^2([0,π])$ subject to periodic, antiperiodic or a general strictly regular boundary condition $(bc)$ has discrete spectrums. It is known that, for large enough $|n|$ in the disc centered at $n$ of radius 1/4, the operator has exactly two (periodic if $n$ is even or antiperiodic if $n$ is odd) eigenvalues $λ_n^+$ and $λ_n^-$ (counted according to multiplicity) and one eigenvalue $μ_n^{bc}$ corresponding to the boundary conditio…
For 1D Dirac operators Ly= i J y' + v y, where J is a diagonal 2x2 matrix with entrees 1,-1 and v(x) is an off-diagonal matrix with L^2 [0,π]-entrees P(x), Q(x) we characterize the class X of pi-periodic potentials v such that: (i) the smoothness of potentials v is determined only by the rate of decay of related spectral gaps gamma (n) = | λ(n,+) - λ(n,-)|, where λ(..) are the eigenvalues of L=L(v) considered on [0,π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions (bc); (ii) there is a Riesz basis which consists of periodic (or antiperiodic) eigenfunctions and (at m…
We prove that there is a homeomorphism between the space of accelerants and the space of potentials of non-self-adjoint Dirac operators on a finite interval.
We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials. In the general non-self-adjoint setting we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb-Thirring inequalities. As physical applications we investigate the damped wave equation and armchair graphene nanoribbons.
Dirac resonances as non-self-adjoint eigenvalues
该研究将三维狄拉克算子的共振定义为预解式亚纯延拓的极点，证明共振集的一个区域可通过畸变狄拉克算子的离散谱刻画且重数不变，并应用该结论在时间反演对称性下证明了克莱默简并性。
Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,π].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \; n\in \mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $γ_n = λ_n^+ - λ_n^-$ and deviations $ …
The Dirac operators $$ Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,π],$$ with $L^2$-potentials $$ v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,π]), $$ considered on $[0,π]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$ S_N = \frac{1}{2πi} \int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \quad P_n = \frac{1}{2πi} \int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz $$ are well--defined for $|n| \geq N$ if $N $ is sufficiently large. It is proved that $$\sum_{|n| > N} \|P_n - P_n^0\|^2 < \infty, $$ where $P_n^…
The one-dimensional Dirac operator \begin{equation*} L = i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{d}{dx} +\begin{pmatrix} 0 & P(x) \\ Q(x) & 0 \end{pmatrix}, \quad P,Q \in L^2 ([0,π]), \end{equation*} considered on $[0,π]$ with periodic and antiperiodic boundary conditions, has discrete spectra. For large enough $|n|,\, n \in \mathbb{Z}, $ there are two (counted with multiplicity) eigenvalues $λ_n^-,λ_n^+ $ (periodic if $n$ is even, or antiperiodic if $n$ is odd) such that $|λ_n^\pm - n |<1/2.$ We study the asymptotics of spectral gaps $γ_n =λ_n^+ - λ_n^-$ in the case $$P(x)=a e^{…
We study the self-adjoint Pauli operators that can be realized as the square of a self-adjoint Dirac operator and correspond to a magnetic field consisting of a finite number of Aharonov-Bohm solenoids and a regular part, and prove an Aharonov-Casher type formula for the number of zero-modes for these operators. We also see that essentially only one of the Pauli operators are spin-flip invariant, and this operator does not have any zero-modes.
Elliptic finite-band potentials of a non-self-adjoint Dirac operator
本文构造了非自伴Dirac算子的双参数椭圆有限带势，连接已知极限情形，通过谱刻画与Heun方程关联，并生成聚焦NLS层次的正负流有限亏格解。
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.
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.
We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrödinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl-Titchmarsh half-line m-coefficients (and 2 x 2 matrix-valued M-matrices) in the non-self-adjoint context and derive some of their basic properties. We conclude with an illustrative example showing that crossing spectral arcs in the non-self-adjoi…
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 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…
This paper is devoted to the mathematical investigation of the MIT bag model, that is the Dirac operator on a smooth and bounded domain with certain boundary conditions. We prove that the operator is self-adjoint and, when the mass goes to infinity, we provide spectral asymptotic results.
We prove the Bari-Markus property for spectral projectors of non-self-adjoint Dirac operators on a finite interval with square-integrable matrix-valued potentials and some separated boundary conditions.
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…
Given an open set $Ω\subset\mathbb{R}^3$. We deal with the spectral study of Dirac operators of the form $H_{a,τ}=H+A_{a,τ}δ_{\partialΩ}$, where $H$ is the free Dirac operator in $\mathbb{R}^3$, $A_{a,τ}$ is a bounded invertible, self-adjoint operator in $\mathit{L}^{2}(\partialΩ)^4$, depending on parameters $(a,τ)\in\mathbb{R}\times\mathbb{R}^n$, $n\geqslant1$. We investigate the self-adjointness and the related spectral properties of $H_{a,τ}$, such as the phenomenon of confinement and the Sobolev regularity of the domain in different situations. Our set of techniques, which is based on fund…
In this note we prove that the maximally defined operator associated with a class of Dirac-type differential expressions M(Q) is J-self-adjoint with respect to a proper antilinear conjugation J under the general hypothesis that the entries of the matrix potential coefficient Q are locally integrable on the real line. The Dirac-type differential expression M(Q) is of significance as it appears in the Lax formulation of the nonabelian (matrix-valued) focusing nonlinear Schrödinger hierarchy of evolution equations.
On the Finite-Zone PT-symmetric Dirac Operators
本文研究了PT对称非自伴Dirac算子，找到了其谱实部能带间隙数目有限的条件。
On the Real Spectum of the One-Dimensional Dirac Operator with PT-Symmetric Coefficients
本文研究具有PT对称周期矩阵值系数的一维非自伴Dirac算子，证明其谱包含实轴的很大一部分。
This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is self-adjoint on a usual Sobolev space whereas when the sector is non-convexit has a family of self-adjoint extensions parametrized by a complex number of theunit circle. As a byproduct of this analysis we are able to give self-adjointnessresults on polygones. We also discuss the question of distinguished self-adjointextensions and study basic spectral propert…
We consider the four-component two-valley Dirac operator on a wedge in $\mathbb{R}^2$ with infinite mass boundary conditions, which enjoy a flip at the vertex. We show that it has deficiency indices $(1,1)$ and we parametrize all its self-adjoint extensions, relying on the fact that the underlying two-component Dirac operator is symmetric with deficiency indices $(0,1)$. The respective defect element is computed explicitly. We observe that there exists no self-adjoint extension, which can be decomposed into an orthogonal sum of two two-component operators. In physics, this effect is called mix…
For one-dimensional Dirac operators $$ Ly= i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{dy}{dx} + v y, \quad v= \begin{pmatrix} 0 & P \\ Q & 0 \end{pmatrix}, \;\; y=\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, $$ subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in $L^2 ([0,π], \mathbb{C}^2).$ In particular, if the potential matrix $v$ is skew-symmetric (i.e., $\overline{Q} =-P$), or more generally if $\overline{Q} =t P$ for some real $t \neq 0,$ then there exists a R…
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.
Spectral enclosure estimates for non-self-adjoint Dirac operators
本文研究了非自伴Dirac算子的谱包络估计，得出了关于谱的精确界及 semiclassical 谱的限制条件。
General $δ$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation
该研究针对二维带闭曲线支撑一般局部奇异相互作用的Dirac算子，将相互作用分解为四类基本相互作用的线性组合开展系统分析，探讨了算子的自伴性、谱性质及正则势Dirac算子的逼近方法。
We prove the absence of eigenvaues of the three-dimensional Dirac operator with non-Hermitian potentials in unbounded regions of the complex plane under smallness conditions on the potentials in Lebesgue spaces. Our sufficient conditions are quantitative and easily checkable.
Semiclassical WKB Problem for the non-self-adjoint Dirac operator
本文研究了非自共轭狄拉克算子的半经典 WKB 问题，探讨了其与聚焦立方 NLS 方程的联系。
This note aims to give prominence to some new results on the absence and localization of eigenvalues for the Dirac and Klein-Gordon operators, starting from known resolvent estimates already established in the literature combined with the renowned Birman-Schwinger principle.
We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.
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.
Depending on the behaviour of the complex-valued electromagnetic potential in the neighbourhood of infinity, pseudomodes of one-dimensional Dirac operators corresponding to large pseudoeigenvalues are constructed. This is a first systematic non-semi-classical approach, which results in substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including superexponential ones.
Non-self-adjoint Dirac operators on graphs
本文研究有限度量图上 Dirac 算子的非自伴实现，利用边界三元组框架推导 Birman-Schwinger 原理变体，并给出算子关于宇称、时间反演和电荷共轭对称的充分必要条件。
We consider Dirac, Pauli and Schrödinger quantum magnetic Hamiltonians of full rank in ${\rm L}^2 \big(\mathbb{R}^{2d} \big)$, $d \ge 1$, perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of non-self-adjoint perturbations, generating near each point of the essential spectrum of the operators, infinitely many (complex) eigenvalues. In particular, we establish point spectrum analogous of Bögli results [Bög17] obtained for non-magnetic Laplacians, and hence showing that classical Lieb-Thirring inequalities cannot hold for our magnetic models. On the …
We show that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis Weyl functions are unique analytic extensions of the reflection coefficients. New results on the extension of the Weyl functions to the real axis and on the existence (in the skew-self-adjoint case) of the Weyl functions follow. Important procedures to recover general-type Dirac systems from the Weyl functions are applied to the recovery of Dirac systems from the reflection coefficients. We explicitly recover Dirac systems from the rational reflection coefficients as well.
Dirac operators with infinite mass boundary conditions on unbounded domains with infinite corners
本文研究带无限质量边界条件的二维Dirac算子在含无限角点的无界区域上的自伴性，证明无凹角点时算子自伴，有凹角点时构造并刻画了特选自伴扩张，并分析了其谱。
We consider the two-dimensional Dirac operator with Lorentz-scalar $δ$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum of half-line Dirac operators with off-diagonal Coulomb potentials. This decomposition reduces the computation of the deficiency indices to determining the number of eigenvalues of a one-dimensional spin-orbit operator in the interval $(-1/2,1/2)$. If the number of edges of the star graph is two or three, these deficiency indices can then be analytically determine…
We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide.
Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices
本文用离散自伴Dirac系统替代Szegő递推与矩阵正交多项式来研究分块Toeplitz矩阵，证明Christoffel–Darboux公式的类似物并导出再生核渐近关系，进而刻画有理Weyl–Titchmarsh函数情形下出现的分块对角加分块半可分离Toeplitz矩阵。
The present paper studies the Dirac Hamiltonian of a test electron with a domain of bi-spinor wave functions supported on the static region inside the Cauchy horizon of the subextremal RWN black hole spacetime, respectively inside the event horizon of the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a sufficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restores essential self-adjointness; the empirical value of the electron's anomalous magnetic…
Self-adjointness for the MIT bag model on an unbounded cone
该研究针对无界三维圆锥上带MIT袋边界条件的无质量Dirac算子，证明凸锥下其在四分量$H^1$函数域上自伴，给出非凸锥情形的数值证据，还得到凸锥上的Hardy型不等式及扰动稳定性，并讨论了向量子点边界条件的推广。
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…
In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Ω; \mathbb{C}^4)$, where $Ω\subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract bound…
Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators
针对兼具块分解与变分间隙性质的无界线性对称算子，提出仅需极弱假设的几何策略，定义带谱间隙的特异自伴延拓并通过极小极大原理刻画间隙内特征值，结果可应用于含变号势或多原子核的类Dirac-Coulomb哈密顿量。
Considering different self-adjoint realisations of positively projected massless Coulomb-Dirac operators we find out, under which conditions any negative perturbation, however small, leads to emergence of negative spectrum. We also prove some weighted Lieb-Thirring estimates for negative eigenvalues of such operators. In the process we find explicit spectral representations for all self-adjoint realisations of massless Coulomb-Dirac operators on the half-line.
The Landau-Dirac operator with shell interactions: self-adjointness and clustering
本文研究带壳相互作用的二维朗道-狄拉克算子，分析其自伴性，证明非临界情形下本征值随曲线对数容量在朗道-狄拉克能级处积累，还得到外边界值问题的本征值聚类结果。
We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime.
The one-dimensional Dirac operator with a singular interaction term which is formally given by $A\otimes|δ_0\rangle\langleδ_0|$, where $A$ is an arbitrary $2\times 2$ matrix and $δ_0$ stands for the Dirac distribution, is introduced as a closed not necessarily self-adjoint operator. We study its spectral properties, find its non-relativistic limit and also address the question of regular approximations. In particular, we show that, contrary to the case of local approximations, for non-local approximating potentials, coupling constants are not renormalized in the limit.
We give the description of self-adjoint regular Dirac operators, on $[0, π]$, with the same spectra.
We study Darboux transformations associated with the focusing nonlinear Schrödinger equation (NLS_-) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J-self-adjoint (but non-self-adjoint) one-dimensional Dirac-type operator with potential q. As one of our principal results we prove that under the most general hypothesis of local integrability on q, the maximally defined operator D(q) generated by this Dirac operator is actually J-self-adjoint. We also establish the existence of Weyl--Titchmarsh-type solutions associated with D(q). The Darboux tra…
On new classes of explicit solutions of Dirac, dynamical Dirac and Dirac--Weyl systems with non-vanishing at infinity potentials, their properties and applications
本文利用GBDT版本的Bäcklund-Darboux变换，构造了具有无穷远处非零位势的自伴与斜自伴Dirac系统、动力学Dirac及Dirac–Weyl系统的广泛新显式解，并推导了Weyl–Titchmarsh函数。
We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the o…
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.
We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate ou…
Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS^+ + A^-S^- + B (with S^\pm the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E_+], E_- < E_+, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D…
Our goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, we consider a couple of Dirac operators.
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