Notes on Rank One Perturbed Resolvent. Perturbation of Isolated Eigenvalue

Eigenvalue and Resonance Asymptotics in perturbed periodically twisted tubes: Twisting versus Bending

We consider the Dirichlet Laplacian in a three-dimensional waveguide that is a small deformation of a periodically twisted tube. The deformation is given by a bending and an additional twisting of the tube, both parametrized by a coupling constant $δ$. We expand the resolvent of the perturbed operator near the bottom of its essential spectrum and we show the existence of exactly one resonance, in the asymptotic regime of $δ$ small. We are able to perform the asymptotic expansion of the resonance in $δ$, which in particular permits us to give a quantitative geometric criterion for the existence…

Discrete spectrum of thin PT-symmetric waveguide

In a thin multidimensional layer we consider a second order differential PT-symmetric operator. The operator is of rather general form and its coefficients are arbitrary functions depending both on slow and fast variables. The PT-symmetry of the operator is ensured by the boundary conditions of Robin type with pure imaginary coefficient. In the work we determine the limiting operator, prove the uniform resolvent convergence of the perturbed operator to the limiting one, and derive the estimates for the rates of convergence. We establish the convergence of the spectrum of perturbed operator to …

Resonances near Thresholds in slightly Twisted Waveguides

We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator, and show the existence of exactly one resonance near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to th…

对称算子扩张的一阶渐近扰动理论

First-order asymptotic perturbation theory for extensions of symmetric operators

本文利用辛几何与Krein公式建立对称算子自伴扩张的一阶渐近扰动理论，推导Riccati方程、特征值变分公式及谱流-Maslov指数关系，并应用于量子图等模型。

The Schrödinger particle on the half-line with an attractive $δ$-interaction: bound states and resonances

In this paper we provide a detailed description of the eigenvalue $ E_{D}(x_0)\leq 0$ (respectively $ E_{N}(x_0)\leq 0$) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution $-λδ(x-x_0)$ for any fixed value of the magnitude of the coupling constant. We also investigate the $λ$-dependence of both eigenvalues for any fixed value of $x_0$. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of t…

Analyticity of Resonances and Eigenvalues and Spectral Properties of the massless Spin-Boson Model

We extend the method of multiscale analysis for resonances introduced in [5] in order to infer analytic properties of resonances and eigenvalues (and their eigenprojections) as well as estimates for the localization of the spectrum of dilated Hamiltonians and norm-bounds for the corresponding resolvent operators, in neighborhoods of resonances and eigenvalues. We apply our method to the massless Spin-Boson model assuming a slight infrared regularization. We prove that the resonance and the ground-state eigenvalue (and their eigenprojections) are analytic with respect to the dilation parameter …

Asymptotics of the eigenvalues of elliptic systems with fast oscillating coefficients

We consider singularly perturbed second order elliptic system in the whole space with fast oscillating coefficients. We construct the complete asymptotic expansions for the eigenvalues converging to the isolated ones of the homogenized system, as well as the complete asymptotic expansions for the associated eigenfunctions.

非正规与亏损矩阵值动力系统的广义特征空间与伪谱

Generalized Eigenspaces and Pseudospectra of Nonnormal and Defective Matrix-Valued Dynamical Systems

本文研究离散时间非正规矩阵动力系统中零特征值的亏损性随时间恢复过程，通过构造广义特征空间和Jordan块分解刻画伪谱的尺寸缩减动力学，并用高斯随机扰动数值实验验证分析有效性。

局部形式从属条件与根系统的Riesz基性

Local form-subordination condition and Riesz basisness of root systems

针对带孤立单正特征值的无界自伴算子非对称扰动问题，引入形式局部从属方法，证明满足间隙与扰动强度条件时根系统含Riesz基且渐近公式成立，并在Schrödinger算子上验证了结果效力。

Resonances in Models of Spin Dependent Point Interactions

In dimension $d=1,2,3$ we define a family of two-channel Hamiltonians obtained as point perturbations of the generator of the free decoupled dynamics. Within the family we choose two Hamiltonians, $\hat H_0$ and $\hat H_\ve$, giving rise respectively to the unperturbed and to the perturbed evolution. The Hamiltonian $\hat H_0$ does not couple the channels and has an eigenvalue embedded in the continuous spectrum. The Hamiltonian $\hat H_\ve$ is a small perturbation, in resolvent sense, of $\hat H_0$ and exhibits a small coupling between the channels. We take advantage of the complete solvabili…

Schrödinger operators with singular rank-two perturbations and point interactions

Norm resolvent approximation for a wide class of point interactions in one dimension is constructed. To analyse the limit behaviour of Schrödinger operators with localized singular rank-two perturbations coupled with δ-like potentials as the support of perturbation shrinks to a point, we show that the set of limit operators is quite rich. Depending on parameters of the perturbation, the limit operators are described by both the connected and separated boundary conditions. In particular an approximation for a four-parametric subfamily of all the connected point interactions is built. We give ex…

弱耦合极限下一维Dirac算子特征值的渐近性

Asymptotics for eigenvalues of one-dimensional Dirac operators in the weak coupling limit

研究弱耦合极限下受扰一维大质量Dirac算子特征值的渐近行为，针对有界Hermite势和L^1势分别得到主导项和渐近展开第二项。