32 research outputs found
On the shape of spectra for non-self-adjoint periodic Schr\"odinger operators
The spectra of the Schr\"odinger operators with periodic potentials are
studied. When the potential is real and periodic, the spectrum consists of at
most countably many line segments (energy bands) on the real line, while when
the potential is complex and periodic, the spectrum consists of at most
countably many analytic arcs in the complex plane.
In some recent papers, such operators with complex -symmetric
periodic potentials are studied. In particular, the authors argued that some
energy bands would appear and disappear under perturbations. Here, we show that
appearance and disappearance of such energy bands imply existence of nonreal
spectra. This is a consequence of a more general result, describing the local
shape of the spectrum.Comment: 5 pages, 2 figure
Biorthogonal Quantum Systems
Models of PT symmetric quantum mechanics provide examples of biorthogonal
quantum systems. The latter incorporporate all the structure of PT symmetric
models, and allow for generalizations, especially in situations where the PT
construction of the dual space fails. The formalism is illustrated by a few
exact results for models of the form H=(p+\nu)^2+\sum_{k>0}\mu_{k}exp(ikx). In
some non-trivial cases, equivalent hermitian theories are obtained and shown to
be very simple: They are just free (chiral) particles. Field theory extensions
are briefly considered.Comment: 34 pages, 5 eps figures; references added and other changes made to
conform to journal versio
Trace Formulas for Schroedinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds
We investigate trace formulas for one-dimensional Schroedinger operators
which are trace class perturbations of quasi-periodic finite-gap operators
using Krein's spectral shift theory. In particular, we establish the conserved
quantities for the solutions of the Korteweg-de Vries hierarchy in this class
and relate them to the reflection coefficients via Abelian integrals on the
underlying hyperelliptic Riemann surface.Comment: 14 page
Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
We give a self-contained presentation of the theory of self-adjoint
extensions using the technique of boundary triples. A description of the
spectra of self-adjoint extensions in terms of the corresponding Krein maps
(Weyl functions) is given. Applications include quantum graphs, point
interactions, hybrid spaces, singular perturbations.Comment: 81 pages, new references added, subsection 1.3 extended, typos
correcte
Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions
© 2020 The Authors. Mathematische Nachrichten published by Wiley‐VCH Verlag GmbH & Co. KGaA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed