Convergent expansions of eigenvalues of the generalized Friedrichs model with a rank-one perturbation

Low energy effects for a family of Friedrichs models

A family of Friedrichs models with rank one perturbations $h_μ(p),$ $p \in (-π,π]^3,μ>0$ associated to a system of two particles on the lattice $\Z^3$ is considered. The existence of a unique strictly positive eigenvalue below the bottom of the essential spectrum of $h_μ(p)$ for all nontrivial values $p \in (-π,π]^3$ under the assumption that $h_μ(0)$ has either a zero energy resonance (virtual level) or a threshold eigenvalue is proved. Low energy asymptotic expansion for the Fredholm determinant associated to family of Friedrichs models is obtained.

The threshold effects for a family of Friedrichs models under rank one perturbations

A family of Friedrichs models under rank one perturbations $h_μ(p),$ $p \in (-π,π]^3$, $μ>0,$ associated to a system of two particles on the three dimensional lattice $\Z^3$ is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of $h_μ(p)$ for all nontrivial values of $p$ under the assumption that $h_μ(0)$ has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.

关于ND中Maxwell和Friedrichs/Poincaré常数的研究

On the Maxwell and Friedrichs/Poincare Constants in ND

本文研究了在任意维度的有界凸域中，Maxwell常数与Friedrichs和Poincaré常数的上下界关系，特别是证明了ND中第二正Maxwell特征值下界由第二Neumann-Laplace特征值的平方根决定。

Spectral perturbation bounds for selfadjoint operators

We give general spectral and eigenvalue perturbation bounds for a selfadjoint operator perturbed in the sense of the pseudo-Friedrichs extension. We also give several generalisations of the aforementioned extension. The spectral bounds for finite eigenvalues are obtained by using analyticity and monotonicity properties (rather than variational principles) and they are general enough to include eigenvalues in gaps of the essential spectrum.

A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions

For an arbitrary open, nonempty, bounded set $Ω\subset \mathbb{R}^n$, $n \in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{Ω, 2m} (a,b,q)$ in $L^2(Ω)$ defined on $W_0^{2m,2}(Ω)$, associated with the higher-order differential expression $$ τ_{2m} (a,b,q) := \bigg(\sum_{j,k=1}^{n} (-i \partial_j - b_j) a_{j,k} (-i \partial_k - b_k)+q\bigg)^m, \quad m \in \mathbb{N}, $$ and its Krein--von Neumann extension $A_{K, Ω, 2m} (a,b,q)$ in $L^2(Ω)$. Denoting by $N(λ; A_{K, Ω, 2m} (a,b,q))$, $λ> 0$, the eigenval…

Threshold phenomenon for a family of the Generalized Generalized Friedrichs models with the perturbation of rank one

A family $H_μ(p),$ $μ>0,$ $p\in\T^3$ of the Generalized Firedrichs models with the perturbation of rank one, associated to a system of two particles, moving on the three dimensional lattice $\mathbb{\Z}^3,$ is considered. The existence or absence of the unique eigenvalue of the operator $H_μ(p)$ lying outside the essential spectrum, depending on the values of $μ>0$ and $p\in U_δ(p_{\,0})\subset\T^3$ is proven. Moreover, the analyticity of associated eigenfunction is shown.