Related papers: On a problem in eigenvalue perturbation theory

Let $A$ be a non-negative self-adjoint operator in a Hilbert space $\mathcal{H}$ and $A_{0}$ be some densely defined closed restriction of $A_{0}$, $A_{0}\subseteq A \neq A_{0}$. It is of interest to know whether $A$ is the unique…

Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $$\rho, b+ic,b-ic, \lambda_4,\cdots,\lambda_n,$$ where $\rho$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a…

Given, on the Hilbert space $\H_0$, the self-adjoint operator $B$ and the skew-adjoint operators $C_1$ and $C_2$, we consider, on the Hilbert space $\H\simeq D(B)\oplus\H_0$, the skew-adjoint operator $$W=[\begin{matrix} C_2&\uno…

We study perturbations of a self-adjoint positive operator $T$, provided that a perturbation operator $B$ satisfies "local" subordinate condition $\|B\varphi_k\|\leqslant b\mu_k^{\beta}$ with some $\beta <1$ and $b>0$. Here…

Let $L_0$ be a bounded operator on a Banach space, and consider a perturbation $L=L_0+K$, where $K$ is compact. This work is concerned with obtaining bounds on the number of eigenvalues of $L$ in subsets of the complement of the essential…

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $\lambda^2 M x + \lambda C x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices…

Let F(u_\ve)+\ve(u_\ve-w)=0 \eqno{(1)} where $F$ is a nonlinear operator in a Hilbert space $H$, $w\in H$ is an element, and $\ve>0$ is a parameter. Assume that $F(y)=0$, and $F'(y)$ is not a boundedly invertible operator. Sufficient…

Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists of irrational or…

We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $\lambda^\circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be "along" the…

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…

We provide sufficient and necessary conditions guaranteeing equations $(A+B)^*=A^*+B^*$ and $(AB)^*=B^*A^*$ concerning densely defined unbounded operators $A,B$ between Hilbert spaces. We also improve the perturbation theory of selfadjoint…

Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union $\bigcup_{j\in J}\Delta_j$, $J\subseteq\mathbb{Z}$, of segments $\Delta_j=[\alpha_j,…

In modeling quantum systems or wave phenomena, one is often interested in identifying eigenstates that approximately carry a specified property; scattering states approximately align with incoming and outgoing traveling waves, for instance,…

We consider a non-self-adjoint $H$ given as the perturbation of a self-adjoint operator $H_0$. We suppose that $H$ is of the form $H=H_0+CWC$ where $C$ is a bounded, positive definite and relatively compact with respect to $H_0$, and $W$ is…

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…

Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral…

We study singular Sturm-Liouville operators of the form \[ \frac{1}{r_j}\left(-\frac{\mathrm d}{\mathrm dx}p_j\frac{\mathrm d}{\mathrm dx}+q_j\right),\qquad j=0,1, \] in $L^2((a,b);r_j)$, where, in contrast to the usual assumptions, the…

We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator $H$. We assume the existence of another self-adjoint operator $A$ for which the family…

We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we…

Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$…