Browsing by Subject "essential spectrum"

Now showing 1 - 4 of 4

Access status: Open Access ,
A note on the discrete Schrödinger operator with a perturbed periodic potential
(Wydawnictwa AGH, 2010) Strack, Beata
The aim of this paper is to study the spectrum of the one-dimensional discrete Schrödinger operator with a perturbed periodic potential. We obtain natural conditions under which this perturbation preserves the essential spectrum of the considered operator. Conditions on the number of isolated eigenvalues are given.
Access status: Open Access ,
Hilderbrand's theorem for the essential spectrum
(2015) Bračič, Janko; Diogo, Cristina
We prove a variant of Hildebrandt’s theorem which asserts that the convex hull of the essential spectrum of an operator $A$ on a complex Hilbert space is equal to the intersection of the essential numerical ranges of operators which are similar to $A$. As a consequence, it is given a necessary and sufficient condition for zero not being in the convex hull of the essential spectrum of $A$.
Access status: Open Access ,
Magnetic Dirichlet Laplacian in curved waveguides
(Wydawnictwa AGH, 2025) Barseghyan, Diana; Bernstein, Swanhild; Schneider, Baruch; Zimmermann, Martha Lina
For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable with respect to waveguide deformations. This means that if the waveguide is a straight strip then the spectrum of the Dirichlet Laplacian is purely essential. From the other hand, the perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, the Dirichlet–Laplace operator with a magnetic field is considered. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small but non-local deformations of the waveguide.
Access status: Open Access ,
On the spectrum of periodic perturbations of certain unbounded Jacobi operators
(2016) Sahbani, Jaouad
It is known that a purely off-diagonal Jacobi operator with coefficients $a_n=n^{\alpha}$, $\alpha\in(0,1]$ has a purely absolutely continuous spectrum filling the whole real axis. We show that a 2-periodic perturbation of these operators creates a non trivial gap in the spectrum.