Related papers: Perturbation of an Eigen-Value from a Dense Point …

We study the fluctuation of the eigenvalue number of any fixed interval $\Delta=[a,b]$ inside the spectrum for $\beta$- ensembles of random matrices in the case $\beta=1,2,4$. We assume that the potential $V$ is polynomial and consider the…

Solvable Hamiltonians for the $\beta$ and $\gamma$ intrinsic shape coordinates are proposed. The eigenfunctions of the $\gamma$ Hamiltonian are spheroidal periodic functions, while the Hamiltonian for the $\beta$ degree of freedom involves…

In ${\cal PT}-$symmetric quantum mechanics one of the most characteristic mathematical features of the formalism is the explicit Hamiltonian-dependence of the physical Hilbert space of states ${\cal H}={\cal H}(H)$. Some of the most…

Eigenvalue and eigenvector perturbation theory is a fundamental topic in several disciplines, including numerical linear algebra, quantum physics, and related fields. The central problem is to understand how the eigenvalues and eigenvectors…

We analyze the Fr\'echet derivative $F$, that maps a perturbation in conductivity to the linearized change in boundary measurements governed by the conductivity equation. The domain is taken to be the unit ball $B \subset \mathbb{R}^d$ with…

In this paper we examine the zero and first order eigenvalue fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles, using the matrix models we described in \cite{dumitriu02}, in the limit as $\beta \to \infty$. We find that…

We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schr{\"o}dinger perturbation series converge near each unperturbed…

Let B=A+K where A is a bounded selfadjoint operator and K is an element of the von Neumann-Schatten ideal S_p with p>1. Let {\lambda_n} denote an enumeration of the discrete spectrum of B. We show that $\sum_n \dist(\lambda_n, \sigma(A))^p$…

In this work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made…

We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(\lambda)=\log(\sigma_k(\lambda))$ on a subset of the positive cone $\Gamma_n=\{\lambda=(\lambda_{1}, \lambda_{2}, \cdots,\lambda_{n})\in \mathbb{R}^n;…

In this paper we adapt the mathematical machinery presented in \cite{P1} to get, by means of the Laplace-Beltrami operator, the discrete spectrum of the Hamiltonian of the Schr\"{o}dinger operator perturbed by an actractive 3D delta…

Recently, several studies of non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry have been conducted. Most striking about these complex Hamiltonians is how closely their properties resemble those of conventional Hermitian…

Let $(\phi_t)_{t \geq 0}$ be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $\tau=1$. The angular derivative is $\phi_t^{\prime}(1)= e^{-\lambda t}$, where $\lambda \geq 0$ is the spectral value…

Some properties of $m$-density points and density-degree functions are studied. Moreover the following main results are provided: \vskip2mm \begin{itemize} \item {\it Let $\lambda$ be a continuous differential form of degree $h$ in…

We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert space $H$ with discrete spectrum $\{t_k \}$, $T \phi_k = t_k \phi_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator…

We discuss operators of the type $H = -\Delta + V(x) - \alpha \delta(x-\Sigma)$ with an attractive interaction, $\alpha>0$, in $L^2(\mathbb{R}^3)$, where $\Sigma$ is an infinite surface, asymptotically planar and smooth outside a compact,…

Matrix elements of observables in eigenstates of generic Hamiltonians are described by the Srednicki ansatz within the eigenstate thermalization hypothesis (ETH). We study a quantum chaotic spin-fermion model in a one-dimensional lattice,…

We consider a generalized Schr\"odinger operator in $L^2(\R^2)$ with an attractive strongly singular interaction of $\delta'$ type characterized by the coupling parameter $\beta>0$ and supported by a $C^4$-smooth closed curve $\Gamma$ of…

Particle decay during inflation is studied by implementing a dynamical renormalization group resummation combined with a small Delta expansion. Delta measures the deviation from the scale invariant power spectrum and regulates the infrared.…

We study the behavior of a quantum particle confined to a hard--wall strip of a constant width in which there is a finite number $ N $ of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein's…