Browsing by Subject "singular continuous spectrum"

Now showing 1 - 2 of 2

Access status: Open Access ,
On spectral stability for rank one singular perturbations
(Wydawnictwa AGH, 2025) Caballero, Mario Alberto Ruiz; Río, Rafael del
We study the embedded point spectrum of rank one singular perturbations of an arbitrary self-adjoint operator $A$ on a Hilbert space $\mathcal{H}$. These perturbations can be regarded as self-adjoint extensions of a densely defined closed symmetric operator $B$ with deficiency indices $(1,1)$. Assuming the deficiency vector of $B$ is cyclic for its self-adjoint extensions, we prove that the spectrum of $A$ contains a dense $G_{\delta}$ subset on which no eigenvalues occur for the rank one singular perturbations considered. We show this is equivalent to the existence of a dense $G_{\delta}$ set of rank one singular perturbations of $A$ such that their eigenvalues are isolated. The approach presented here unifies points of view taken by different authors.
Access status: Open Access ,
Singular continuous spectrum of half-line Schrödinger operators with point interactions on a sparse set
(2011) Lotoreichik, Vladimir
We say that a discrete set $X = \{ x_n \}_{n\in \mathbb{N}_0}$ on the half-line $0 = x_0 \lt x_1 \lt x_2 \lt x_3 \lt ... \lt x_n \lt ... \lt +\infty$ is sparse if the distances $\Delta x_n = x_{n+1}- x_n$ between neighbouring points satisfy the condition $\frac{\Delta x_n}{\Delta x_{n-1}} \to +\infty$. In this paper half-line Schrödinger operators with point $\delta#- and $\delta'$- interactions on a sparse set are considered. Assuming that strengths of point interactions tend to $\infty$ we give simple sufficient conditions for such Schrödinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with $\mathbb{R}_+$.