Perturbation Theory of Resonances and Embedded Eigenvalues of the Schrodinger Operator For a Crystal Film

We obtain formulas for resonances and eigenvalues embedded in the continuous spectrum that are similar to formulas in the standard perturbation theory. We prove that although the imaginary part of the first-order correction to the eigenvalue embedded in the continuous spectrum is zero, the perturbed eigenfunction, as a rule, ceases to be square-summable.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 417–430, June, 2005.     

双重内共振系统非线性模态分岔的奇异性分析

利用多尺度法构造的一类1:2:5双重内共振系统的耦合非线性模态的分岔是一个两变量的分岔问题.利用Maple计算机代数可以通过消元将耦合的模态分岔方程分离为两个单变量的分岔方程.对分离后的单变量分岔方程进行奇异性分析,发现随着系统参数的变化,非线性模态的分岔既可以是一种模态向另一种模态的转化,也可以是一种模态的突然出现与消失.最后给出了两变量分岔问题可以利用消元后得到的单变量分岔方程和耦合方程进行处理的一种方法.     

Resonances and tunneling in a quantum wire

We consider a problem in mathematical scattering theory related to the ballistic conductance model. The model under investigation describes the charge propagation in a quantum wire. We assume that the charge carrier has a spin and take the Rashba spin-orbital interaction into account. We study the conductance resonances generated by the weak quantum-wire interaction with the quasistationary state of a parallel-connected quantum dot or with the tunneling through a series-connected quantum dot. Such a quantum dot is usually the control element. We present sufficient conditions for the spatial symmetry of the system to ensure that the quasistationary state of the quantum dot generates a conductance resonance. We assume that the conductance is related to the scattering matrix by the Landauer formula. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 92–102, April, 2006.     

一类线性非自伴算子的谱

近年来国际上出现了一类以迁移理论为背景的所谓＂ｕ－标算子（ｕ－ｓｃａｌａｒｏｐｅｒａｔｏｒ）＂．本文研究ｕ－标算子与经典的标型谱算子（ｓｐｅｃｔｒａｌｏｐｅｒａｔｏｒｏｆｓｃａｌａｒｔｙｐｅ）的内在联系，给出了此类算子的谱与其无界投影族之间的关系，证明了该类算子的剩余谱是空集．     

Operator Interpretation of Resonances Arising in Spectral Problems for 2 × 2 Operator Matrices

We consider operator matrices ${\rm{H = }}\left( {{_{B_{10} }^{A{_0}}} {_{A_{1} }^{B{_{01}}}} } \right)$ with self - adjoint entries A_i, i = 0,1, and bounded B₁₀ = B₁₀ acting in the orthogonal sum μ=μ₀ ⊕ μ₁ of Hilbert spaces μ₀ and μ₁. We are especially interested in the case where the spectrum of, say, A₁ is partly or totally embedded into the continuous spectrum of A₀ and the transfer function V₁(z) = B₁₀ where (χ - A₀)^?1 B₀₁, admits analytic continuation (as an operator - valued function) through the cuts along branches of the continuous spectrum of the entry A₀ into the unphysical sheet(s) of the spectral parameter plane. The values of χ in the unphysical sheets where M₁^?1(z) and consequently the resolvent (H - z)^?1 have poles are usually called resonances. A main goal of the present work is to find non - selfadjoint operators whose spectra include the resonances as well as to study the completeness and basis properties of the resonance eigenvectors of M₁(z) in μ₁. To this end we first construct an operator - valued function V₁(Y) on the space of operators in μ₁ possessing the property: V₁(Y)?₁ = V₁(z)?₁ for any eigenvector ?₁ of Y corresponding to an eigenvalue z and then study the equation H₁ = A₁ + V₁(H₁). We prove the solvability of this equation even in the case where the spectra of A₀ and A₁ overlap. Using the fact that the root vectors of the solutions H₁ are at the same time such vectors for M₁(z), we prove completeness and even basis properties for the root vectors (including those for the resonances).     

Mathematical Model of Resonances and Tunneling in a System with a Bound State

We study the asymptotic behavior of the residue at the pole of the analytic continuation of the scattering matrix as the imaginary part of the pole tends to zero in the case where the phase space of a quantum mechanical system is a direct sum of two spaces and the nonperturbed evolution operator reduces each of these spaces and has a discrete spectrum in one of them and a continuous spectrum in the other. The perturbation operator mixes the subspaces and generates a resonance. We prove that under certain symmetry conditions in such a system, the scattering amplitude changes sharply in a neighborhood of the real part of the pole of the scattering matrix, and the system demonstrates tunneling or a resonance of the scattering amplitude.     

Counting Function of Characteristic Values and Magnetic Resonances

We consider the meromorphic operator-valued function I ? K(z) = I ? A(z)/z where A is holomorphic on the domain 𝒟 ? ?, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic values of I ? K, i.e. the complex numbers w ≠ 0 for which the operator I ? K(w) is not invertible, and we show that generically the characteristic values of I ? K converge to 0 with the same rate as the eigenvalues of A(0).

We apply our abstract results to the investigation of the resonances of the operator H = H ₀ + V where H ₀ is the shifted 3D Schrödinger operator with constant magnetic field of scalar intensity b > 0, and V: ?³ → ? is the electric potential which admits a suitable decay at infinity. It is well known that the spectrum σ(H ₀) of H ₀ is purely absolutely continuous, coincides with [0, + ∞[, and the so-called Landau levels 2bq with integer q ≥ 0, play the role of thresholds in σ(H ₀). We study the asymptotic distribution of the resonances near any given Landau level, and under generic assumptions obtain the main asymptotic term of the corresponding resonance counting function, written explicitly in the terms of appropriate Toeplitz operators.     

J—自共轭微分算子谱的定性分析

本文对J－自共轭微分算子谱理论研究情况做一些概要性的介绍,第一部分简要回顾了J－自共轭微分算子理论研究的发展过程,第二,三部分介绍了J－自共轭微分算子的本质谱和离散谱定性分析的主要方法和结论;第四部分扼要叙述J－自共轭微分算子其它方面的一些工作,以及J－自共轭微分算子谱理论研究中尚待解决的问题。     

一类自伴微分算子谱的离散性

本文研究了２ｎ阶实系数Ｅｕｌｅｒ微分算式生成的对称微分算子,得到了自伴Ｅｕｌｅｒ微分算子的谱是离散的充分必要条件．     

上三角无穷维Hamilton算子的剩余谱

将点谱划分为四个部分,得到上三角无穷维Hamilton算子的点谱σ_p(H)关于虚轴对称的充要条件.在此基础上,结合无穷维Hamilton算子的谱结构,得到无穷维Hamilton算子剩余谱的完全描述,从而实现了利用其内部算子刻画剩余谱.     

L2×L2中的一类无穷维Hamilton算子的剩余谱

该文得到了一类无穷维Hamilton算子的剩余谱和点谱存在的几个判别准则,从而给出 了求其剩余谱和点谱的方法．在此基础上构造了L2×L2中无穷维Hamilton算子的剩余谱 非空的具体例子,从而进一步验证了判别准则的有效性．     

一类无穷维Hamilton算子的谱

研究了一类无穷维Hamilton算子的近似点谱及本质谱.进而通过无穷维Hamilton算子内部元素的乘积的谱对整体谱进行了刻画,最后证明了结论的正确性.     

一类缺项算子矩阵的谱扰动

有界线性算子的点谱和剩余谱分别可进-步细分为两类:σ_(p1),σ_(p2)和σ_(r1),σ_(r2).设H,K为无穷维可分的Hilbert空间,本文将对于给定的A ∈B (H),B ∈B(K),给出了缺项算子M_C=(AC/OB)关于分类后所得四种谱的扰动结果.     

一类具指数系数的对称微分算子谱的离散性

研究了具指数函数系数的2n阶实系数微分算式生成的对称微分算子,利用算子的直和分解法及不等式估计得到此类微分算子谱是离散的充分条件.     

偶数阶自共轭微分算子谱的离散性条件

In this paper, we consider differential operators of 2nd-order
a[u]=(-1)^k(a_k(x)u^(k)(x))(k), x∈(0, ∞)
whose coefficients a_k(x) are restricted by powers of e^x, and give conditions on the coefficients sufficient to ensure that the spectrum is discrete; next we formulate necessary and sufficient conditions for the discreteness of the spectrum of differential operators whose coefficients a_k(x) may increase as e^αkx as x→∞.     

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as (0.1)but the last term on the right of (0.1) is replaced by a relatively form‐compact perturbation of a similar form. We study relations between the operator function T, a selfadjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of the operator function –T^–1. The results are applied to a Sturm‐Liouville problem with a coefficient depending rationally on the eigenvalue parameter.     

复系数2n阶微分算子的谱

本文研究了复系数２ｎ阶微分算式（２．１）生成的Ｊ－自伴微分算子谱,对两类微分算子的本质谱,离散谱作了定性研究,得到了所生成微分算子本质谱的存在范围,以及所生成微分算子的谱是离散的充分条件．     

Resonances and Tunneling in the Tight-Binding Approximation to Scattering in a Quantum Billiard

In the tight-binding approximation, we consider the scattering problem in a quantum billiard and investigate the behavior of the scattering matrix near a resonance.