Fatma Karray Meziou
Let $u=\int_{-\infty}^{+\infty}λdE_λ$ be a self-adjoint operator in a Hilbert space $H$. Our purpose is to provide a non-standard description of the spectral family $(E_λ)$ and the generalized Gelfand eigenvectors.
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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 $L^p([0,π]; dx)$, $p \in (1,\infty)$.
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.
Let $X$ be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra $H$ of $C_B(X)$ which has local units we construct the spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the Stone-Cech compactification of $X$ which contains $X$ as a dense subspace. The construction of $\mathfrak{sp}(H)$ is simple. This enables us to study certain properties of $\mathfrak{sp}(H)$, among them are various compactness and connectedness properties. In particular, we find necessary and sufficient conditions in terms of either $H$ or $X$ under which $\mathfrak{sp}(H)$ is connected, lo…
We perform the spectral analysis of a family of Jacobi operators $J(α)$ depending on a complex parameter $α$. If $|α|\neq1$ the spectrum of $J(α)$ is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If $|α|=1$, $α\neq \pm 1$, the essential spectrum of $J(α)$ covers the entire complex plane. In addition, a formula for the Weyl $m$-function as well as the asymptotic expansions of solutions of the difference equation corresponding to $J(α)$ are obtained. Finally, the completeness of eigenvectors and Rodriguez-li…
Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a compact group of automorphisms of N, and the algebra D(N)^K of left-invariant and K-invariant differential operators on N is commutative. In these hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies Vinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N). Fixing a system D of d formally self-adjoint generators of D(N)^K, the Gelfand spectrum of the commutative convolution algebra L^1(N)^K can be canonically identified with a closed subset S_D of R^d. We prove that, on a nilpot…
Let $H$ be a self-adjoint operator, bounded from below and let $O$ be a bounded self-adjoint operator with purely discrete spectrum. Suppose that (i) $E(H)=\inf \mathrm{spec}(H)$ is a simple eigenvalue, and (ii) $H$ strongly commutes with $O$. Let $ψ_H$ be the eigenvector associated with $E(H)$. By the assumptions (i) and (ii), $ψ_H$ is an eigenvector of $O$: $Oψ_H=μ(H)ψ_H$. In the context of quantum mechanics, $μ(H)$ is called a good quantum number. In this note, we examine the stability of $μ(H)$ under perturbations of $H$ from a viewpoint of the order theory.
In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution. We then show that the supercell method for perturbed periodic Schrödinger operators falls into the scope of our study. We prove that this method is spectral pollution free, and we derive optimal convergence rates for the planewave discretization meth…
In [U] (among other results), M. Uchiyama gave the necessary and sufficient conditions for contractions to be similar to the unilateral shift $S$ of multiplicity $1$ in terms of norm-estimates of complete analytic families of eigenvectors of their adjoints. In [G2], it was shown that this result for contractions can't be extended to power bounded operators. Namely, a cyclic power bounded operator was constructed which has the requested norm-estimates, is a quasiaffine transform of $S$, but is not quasisimilar to $S$. In this paper, it is shown that the additional assumption on a power bounded …
For a completely regular space $X$ and a non-vanishing self-adjoint closed subalgebra $H$ of $C_B(X)$ which separates points from closed sets in $X$ we construct the Gelfand spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the compactification of $X$ generated by $H$. The simple construction of $\mathfrak{sp}(H)$ enables easier examination of its properties. We illustrate this by an example showing that the space $\mathfrak{sp}(H)$ is separable metrizable if and only if $H$ is countably generated.
M. Krein proved in 1948 that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T* has an eigenvector. We present generalizations of this result as well as some applications to C*-algebras, operators on l_1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras.
Positive eigenvectors and simple nonlinear maps
本文刻画了 Banach 与 Hilbert 空间中至多二对一的两类简单非线性映射，统一并推广了 Ambrosetti-Prodi 定理等经典例子，并借助基态非退化性与 Krein-Rutman 定理给出新例。
Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for $C^\om$ (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a Hölder continuity class $C^{0,\al}$. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.
On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry
本文研究了紧黎曼流形上拉普拉斯-贝尔特里米算子的自共轭扩展构造及二次型的作用，提出了一种计算任意维度特征值和特征向量的数值方案，并探讨了对称算子下不变性的角色。
The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a "weighted subcoercive operator" of ter Elst and Robinson (J. Funct. Anal. 157 (1998) 88-163). The joint spectrum of L_1,...,L_n in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L_1,...,L_n. Connections with the theory of Gelfand pairs are est…
The Bargmann-Fock-Segal space plays an important role in mathematical physics, and has been extended into a number of directions. In the present paper we imbed this space into a Gelfand triple. The spaces forming the Fréchet part (i.e. the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and G. Salomon.
Bounds for the Rayleigh quotient and the spectrum of self-adjoint operators
本文通过残量范数和向量变化量给出Rayleigh商的绝对变化界，并提出统一方法证明谱的已知界、分析其尖锐性并推导新的更紧的界。
The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of this operator and its 'modular partner', based on an in-depth analysis of the eigenvectors of a Hilbert-Schmidt integral operator whose integral kernel has a previously known relation to the two difference operators. With suitable restrictions on the parameters, we show that the commuting difference operators …
Numerical Solutions of the spectral problem for arbitrary self-adjoint extensions of the 1D Schroedinger equation
提出一种数值算法，用于求解一维正则薛定谔算子任意自伴延拓的谱问题，该算法基于非局部边界有限元方法，并证明了收敛性，通过数值实验验证了边界条件扰动导致负基态绝对值增大和边界态局域化。
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。
An Inverse Theorem for the Perron--Frobenius Theorem
本文研究Perron--Frobenius定理的逆问题，证明了当有界正自伴算子的最大特征值为单重时存在使其遍历的希尔伯特锥，分析了专用希尔伯特锥并给出磁Schrödinger算子热半群的应用示例。
A Characterization of Complex Symmetric Weighted Composition Operators on the Fock Space
该研究未预先指定共轭算子，借助再生核与生成函数刻画Fock空间上有界复对称加权复合算子，所得共轭算子多为反线性高斯积分算子，部分为加权复合共轭算子。
We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated. A parallel spectral theory is developed for one-dimensional singular perturbations …
In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, conditions for them to constitute a "continuous basis" for the smallest space $\mathcal D$ of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frame, Riesz basis and orthonormal basis. A motivation for this study comes from the Gel'fand-Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain …
This thesis is devoted to the study of joint spectral multipliers for a system of pairwise commuting, self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G. Under the assumption that the algebra generated by L_1,...,L_n contains a weighted subcoercive operator - a notion due to ter Elst and Robinson (J. Funct. Anal., 157(1):88--163, 1998), including positive elliptic operators, sublaplacians and Rockland operators - we prove that L_1,...,L_n are (essentially) self-adjoint and strongly commuting on L^2(G). Moreover, we perform an abstract study of such a syst…
Let $E \ni x\mapsto A(x)$ be a $\mathscr{C}$-mapping with values unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr{C}$ stands for $C^\infty$, $C^ω$ (real analytic), $C^{[M]}$ (Denjoy--Carleman of Beurling or Roumieu type), $C^{0,1}$ (locally Lipschitz), or $C^{k,α}$. The parameter domain $E$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We completely describe the $\mathscr{C}$-dependence on $x$ of the eigenvalues and the eigenvectors of $A(x)$. Thereby we extend previously known results for self-adjoin…
We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space $X$, one can construct a $T$-invariant probability measure which contains $x_0$ in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrenc…
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