Alexander Pushnitski
Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference $φ(H)-φ(H_0)$ for piecewise continuous functions $φ$. This description involves the scattering matrix for the pair $H$, $H_0$.
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In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,λ)$. The purpose of the paper is to study the spectral properties of the difference $D(λ)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(λ)$ in terms of the eigenvalues of the scattering matrix $S(λ)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(λ)$ is empty.
This paper deals with the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential $q$ is known a priori on a subinterval $ \left[ b,π\right] $ with $b\in \left( d,π\right] $ or $b=d$, then $h,$ $β,$ $γ\ $and $q$ on $\left[ 0,π\right] \ $can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the cas…
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 self-adjoint operators $A, B$, a bounded operator $J$, and a function $f:\mathbb R\to\mathbb C$ we obtain bounds in quasi-normed ideals of compact operators for the difference $f(A)J-Jf(B)$ in terms of the operator $AJ-JB$. The focus is on functions $f$ that are smooth everywhere except for finitely many points. A typical example is the function $f(t) = |t|^γ$ with $γ\in (0, 1)$. The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace $tr f(S)$ with $S$ being a Wiener-Hopf operator with a discontinuous symbol.
On the spectra of periodic discontinuous quantum graphs
本文研究六角格上带自伴不连续顶点条件的周期薛定谔算子，推广石墨烯量子图模型，给出色散关系、谱及狄拉克点，并证明Borg定理不成立。
Essential Self-Adjointness of the Geometric Deformation Operator on a Compact Interval
本文在紧区间上定义了由光滑函数构造的二阶微分算子，通过验证对称性和计算为零的亏指数证明其本质自伴性，为后续谱分析奠定基础。
Let $G$ be a Métivier group and let $N$ be any homogeneous norm on $G$. For $α>0$ denote by $w_α$ the function $e^{-N^α}$ and consider the weighted sub-Laplacian $\mathcal{L}^{w_α}$ associated with the Dirichlet form $ϕ\mapsto \int_{G} |\nabla_\mathcal{H}ϕ(y)|^2 w_α(y)\, dy$, where $\nabla_\mathcal{H}$ is the horizontal gradient on $G$. Consider $\mathcal{L}^{w_α}$ with domain $C_c^\infty$. We prove that $\mathcal{L}^{w_α}$ is essentially self-adjoint when $α\geq 1$. For a particular $N$, which is the norm appearing in $\mathcal{L}$'s fundamental solution when $G$ is an H-type group, we prove …
The inverse problem for the differential operator pencil with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem
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…
Let~$H_0$ and~$V$ be self-adjoint operators such that~$V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact~$F.$ Flow of singular spectrum of the path of self-adjoint operators $H_0 + rV,$ $r \in \mathbb R,$ -- also called spectral flow, through a point $λ$ outside the essential spectrum of~$H_0$ is well studied, and appears in such diverse areas as differential geometry and condensed matter physics. Inside the essential spectrum the spectral flow through $λ$ for such a path is well-defined if the norm limit $$ \lim_{y \to 0^+} F (H_0 + r V - λ- iy)^{-1…
We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.
We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on the space of smooth functions with compact support, and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli-Dirac operators that describe the energy of Hydrogen type atoms on Riemannian 3-manifo…
We consider perturbed quadharmonic operators, $Δ^4 + V$, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential $V$ satisfying a bound from below by a non-positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vector bundle and the underlying Riemannian manifold, we give a sufficient condition for the essential self-adjointness of such operators. We then apply this to prove the separation property in $L^2$ when the perturbed operator acts on functions.
On creating new essential spectrum by self-adjoint extension of gapped operators
本文针对无限亏指数的稠定有间隙对称算子，给出一般延拓框架以构造使间隙任意指定部分成为本质谱的自伴延拓，并阐明其简洁的算子论机制。
We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We give necessary conditions for this Laplacian to be sectorial. We introduce a special self-adjoint operator and compare its essential spectrum with that of the non self-adjoint Laplacian considered.
We give sufficient conditions for essential self-adjointness of magnetic Schrödinger operators on locally finite graphs. Two of the main theorems of the present paper generalize recent results of Torki-Hamza.
We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green's function and when it gives a non-constant harmonic function which is square integrable.
Given a bounded smooth domain $Ω\subset\mathbb{R}^3$, we explore the relation between couplings of the free Dirac operator $-iα\cdot\nabla+mβ$ with pure electrostatic shell potentials $λδ_{\partialΩ}$ ($λ\in\mathbb{R}$) and some perturbations of those potentials given by the normal vector field $N$ on the shell $\partialΩ$, namely $\{λ_e+λ_n(α\cdot N)\}δ_{\partialΩ}$ ($λ_e$, $λ_n\in\mathbb{R}$). Under the appropiate change of parameters, the couplings with perturbed and unperturbed electrostatic shell potentials yield unitary equivalent self-adjoint operators. The proof relies on the construct…
We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrödinger operators in the metrically non complete case.
Essential self-adjointness of non-semibounded Schrödinger operators on infinite graphs
本文研究无穷加权图上未必下半有界的离散薛定谔算子的本质自伴性，给出了相关充分条件，并推导得到势函数满足特定二次下界时算子本质自伴的推论。
We define the magnetic Schrödinger on an infinite graph by the data of a magnetic field, some weights on vertices and some weights on edges . We discuss essential self-adjointness of this operator for graphs of bounded degree. The main result is a discrete version of a result of two authors of the present paper.
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner--Wienholtz--Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.
The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space $L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ induced by matrix differential expressions of the form \begin{align}\label{abstract:mdo} \left(\begin{array}{cc} τ_{11}(\,\cdot\,,D) & τ_{12}(\,\cdot\,,D)\\[3.5ex] τ_{21}(\,\cdot\,,D) & τ_{22}(\,\cdot\,,D) \end{array}\right), \end{align} where $τ_{11}$, $τ_{12}$, $τ_{21}$, $τ_{22}$ are respectively $m$-th, $n$-th, $k$-th and 0 order ordinary differential expressions with $m=n+k$ being even. Under suitable assumption…
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph. Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the weighted graph Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us …
The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. This part is based on joint work with Mark Malamud…
Let $M$ be a strongly pseudoconvex complex manifold which is also the total space of a principal $G$-bundle with $G$ a Lie group and compact orbit space $\bar M/G$. Here we investigate the $\bar\partial$-Neumann Laplacian on $M$. We show that it is essentially self-adjoint on its restriction to compactly supported smooth forms. Moreover we relate its spectrum to the existence of generalized eigenforms: an energy belongs to $σ(\square)$ if there is a subexponentially bounded generalized eigenform for this energy. Vice versa, there is an expansion in terms of these well-behaved eigenforms so tha…
Let $A(t)$ be a holomorphic family of self-adjoint operators of type (B) on a complex Hilbert space $\mathcal{H}$. Kato-Rellich perturbation theory says that isolated eigenvalues of $A(t)$ will be analytic functions of $t$ as long as they remain below the minimum of the essential spectrum of $A(t)$. At a threshold value $t_0$ where one of these eigenvalue functions hits the essential spectrum, the corresponding point in the essential spectrum might or might not be an eigenvalue of $A(t_0)$. Our results generalize a theorem of Simon to give a sufficient condition for the minimum of the essentia…
Diagonals of self-adjoint operators II: Non-compact operators
研究可分无限维Hilbert空间上自伴算子所有可能对角线的刻画；对本质谱至少含两点的算子，在谱测度相同且端点特征值重数可例外的情形给出完整刻画，并对固定算子及特殊本质谱端点情形作更精细描述，推广了Kadison等人的结果。
A famous theorem due to Weyl and von Neumann asserts that two bounded self-adjoint operators are unitarily equivalent modulo the compacts, if and only if their essential spectrum agree. The above theorem does not hold for unbounded operators. Nevertheless, there exist closed subsets $M$ of $\mathbb{R}$ on which the Weyl--von Neumann Theorem hold: all (not necessarily bounded) self-adjoint operators with essential spectrum $M$ are unitarily equivalent modulo the compacts. In this paper, we determine exactly which $M$ satisfies this property.
It is well known that the standard projection methods allow one to recover the whole spectrum of a bounded self-adjoint operator but they often lead to spectral pollution, i.e. to spurious eigenvalues lying in the gaps of the essential spectrum. Methods using second order relative spectra are free from this problem, but they have not been proven to approximate the whole spectrum. L. Boulton (2006, 2007) has shown that second order relative spectra approximate all isolated eigenvalues of finite multiplicity. The main result of the present paper is that second order relative spectra do not in ge…
Using the concept of intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree.
Let $Ω$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-Δ+V(x)$ be a Schrödinger operator on $Ω$ with potential V locally bounded. We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on $C_0^\infty(Ω)$. As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition $V(x)\geq \frac{3}{4d(x)^2}$, where $d(x)=dist(x,\partialΩ)$. The constant 1 in front of each logarithmic term in Theorem 2 is optimal. The proof…
We use C*-algebra theory to provide a new method of decomposing the eseential spectra of self-adjoint and non-self-adjoint Schrodinger operators in one or more space dimensions.
In this paper we construct the spectral expansion for the non-self-adjoint differential operators generated in the space of vektor functions by the ordinary differential expression of arbitrary order with the periodic matrix coefficients by using the essential spectral singularities, singular quasimomenta and the series with parenthesis.
We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms of completeness of certain metrics on M; these metrics are naturally associated to the operator. We do not assume a priori that M is endowed with a complete Riemannian metric. This allows us to treat e.g. operators acting on bounded domains in the euclidean space. For the case when the principal symbol of the operator is scalar, we establish more precise res…
We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We analyse spectral properties of this Laplacian under a Kirchhoff assumption. Moreover we establish isoperimet-ric inequalities in terms of the numerical range to show the absence of the essential spectrum of the Laplacian on heavy end directed graphs.
We study $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$.
Quantum Systems with jump-discontinuous mass. I
研究一维自由量子粒子在质量剖面存在跃变不连续时的谱问题，分析边界条件对谱特性的影响，发现能量依赖性极强且存在无限多个 semiclassical 极限。
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.
We study the question of magnetic confinement of quantum particles on the unit disk $\ID$ in $\IR^2$, i.e. we wish to achieve confinement solely by means of the growth of the magnetic field $B(\vec x)$ near the boundary of the disk. In the spinless case we show that $B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}$, for $|\vec x|$ close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants $\frac{\sqrt 3}{2}$ and $-\frac{1}{\sqrt …
In this paper we construct the spectral expansion for the differential operator generated in all real line by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.
The essential self-adjointness of the wave operator on radiative spacetimes
本文证明辐射时空中达朗贝尔算子的本质自伴性，涵盖引力辐射扰动的闵可夫斯基时空，利用微局部方法和Taira的分布正则性结果。
The notions of magnetic difference operator defined on weighted graphs or magnetic exterior derivative are discrete analogues of the notionof covariant derivative on sections of a fibre bundle and its extension on differential forms. In this paper, we extend this notion to certain 2-simplicial complexes called triangulations, in a manner compatible with changes of gauge. Then we study the magnetic Gauss-Bonnet operator naturally defined in this context and introduce the geometric hypothesis of $χ-$completeness which ensures the essential self-adjointness of this operator. This gives also the e…
This paper is a continuation of the study of spectral flow inside essential spectrum initiated in \cite{AzSFIES}. Given a point $λ$ outside the essential spectrum of a self-adjoint operator $H_0,$ the resonance set, $\mathcal R(λ),$ is an analytic variety which consists of self-adjoint relatively compact perturbations $H_0+V$ of $H_0,$ for which $λ$ is an eigenvalue. One may ask for criteria for the vector $V$ to be tangent to the resonance set. Such criteria were given in \cite{AzSFnRI}. In this paper we study similar criteria for the case of $λ$ inside the essential spectrum of $H_0.$ For th…
In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenv…
In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.
For a very general class of unbounded self-adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum. These upper bounds are given by triple variations. Furthermore, we find conditions which imply that a point is in the resolvent set. For norm resolvent continuous operator functions we show that the variational inequality becomes an equality.
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 …
Essentially singular limits of Jacobi operators and applications to higher-order squeezing
本文研究了Jacobi算子在耦合参数λ≥0下的本质奇异极限，通过分析小λ情形下的广义特征向量统一界，证明了在λ→0时算子在强解析意义下的收敛性，并应用该结果探讨了量子光学中高阶压缩算子的自伴随扩展问题。
We propose a general condition, to ensure essential self-adjointness for the Gauß-Bonnet operator, based on a notion of completeness as Chernoff. This gives essential self-adjointness of the Laplace operator both for functions or 1-forms on infinite graphs. This is used to extend Flanders result concerning solutions of Kirchhoff's laws.
Essential self-adjointness of strongly singular homogeneous polyharmonic operators
研究强奇异齐次多项调和算子在 $ L^2 $ 空间中的本质自 adjoint 性，推导了不同维度下算子自 adjoint 的临界常数条件。
Let $Ω_-$ and $Ω_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $Σ$. For $μ>0$, consider the function $h_μ=1_{Ω_-}-μ1_{Ω_+}$. We discuss self-adjoint realizations of the operator $L_μ=-\nabla\cdot h_μ\nabla$ in $L^2(Ω_-\cupΩ_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_μ$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Σ$) and study some properties of its unique self-adjoint extension $\mathcal{L}_μ:=\overline{L_μ}$. If $μ\ne 1$, then $\mat…
In this paper we investigate the non-self-adjoint operator H generated in all real line by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.
We discuss the essential spectrum of essentially self-adjoint elliptic differential operators of first order and of Laplace type operators on Riemannian vector bundles over geometrically finite orbifolds.
In this note, we study existence of the outgoing/incoming resolvents of repulsive Schrödinger operators which may not be essentially self-adjoint on the Schwartz space. Moreover, we recover the classical result: The repulsive Schroödinger operators with large repulsive constant is not essentially self-adjoint on the Schwartz space.
We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates. These results extend, and improve, classical perturbation results by Kato and by Gohberg/Krein. Further, we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator; in the latter case, we can even allow for perturbations with relative bound $\ge 1$. The generality of our results is…
On essential-selfadjointness of differential operators on closed manifolds
本文围绕闭流形上形式自伴微分算子的本质自伴性提出猜想，认为其等价于算子象征的哈密顿流完备，并验证了该猜想在几类微分算子上的成立性。
For a Riemannian covering $p \colon M_{2} \to M_{1}$, we compare the spectrum of an essentially self-adjoint differential operator $D_{1}$ on a bundle $E_{1} \to M_{1}$ with the spectrum of its lift $D_{2}$ on $p^{*}E_{1} \to M_{2}$. We prove that if the covering is infinite sheeted and amenable, then the spectrum of $D_{1}$ is contained in the essential spectrum of any self-adjoint extension of $D_{2}$. We show that if the deck transformations group of the covering is infinite and $D_{2}$ is essentially self-adjoint (or symmetric and bounded from below), then $D_{2}$ (or the Friedrichs extens…
In this article we prove a generalization of Weyl's criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.
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