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 $\mathcal{PT}$-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

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

Adipose atrophy in cancer cachexia:morphologic and molecular analysis of adipose tissue in tumour-bearing mice

Extensive loss of adipose tissue is a hallmark of cancer cachexia but the cellular and molecular basis remains unclear. This study has examined morphologic and molecular characteristics of white adipose tissue in mice bearing a cachexia-inducing tumour, MAC16. Adipose tissue from tumour-bearing mice contained shrunken adipocytes that were heterogeneous in size. Increased fibrosis was evident by strong collagen-fibril staining in the tissue matrix. Ultrastructure of 'slimmed' adipocytes revealed severe delipidation and modifications in cell membrane conformation. There were major reductions in mRNA levels of adipogenic transcription factors including CCAAT/enhancer binding protein alpha (C/EBPα), CCAAT/enhancer binding protein beta, peroxisome proliferator-activated receptor gamma, and sterol regulatory element binding protein-1c (SREBP-1c) in adipose tissue, which was accompanied by reduced protein content of C/EBPα and SREBP-1. mRNA levels of SREBP-1c targets, fatty acid synthase, acetyl CoA carboxylase, stearoyl CoA desaturase 1 and glycerol-3-phosphate acyl transferase, also fell as did glucose transporter-4 and leptin. In contrast, mRNA levels of peroxisome proliferators-activated receptor gamma coactivator-1alpha and uncoupling protein-2 were increased in white fat of tumour-bearing mice. These results suggest that the tumour-induced impairment in the formation and lipid storing capacity of adipose tissue occurs in mice with cancer cachexia. © 2006 Cancer Research UK

Genomic Restructuring in the Tasmanian Devil Facial Tumour: Chromosome Painting and Gene Mapping Provide Clues to Evolution of a Transmissible Tumour

Devil facial tumour disease (DFTD) is a fatal, transmissible malignancy that threatens the world's largest marsupial carnivore, the Tasmanian devil, with extinction. First recognised in 1996, DFTD has had a catastrophic effect on wild devil numbers, and intense research efforts to understand and contain the disease have since demonstrated that the tumour is a clonal cell line transmitted by allograft. We used chromosome painting and gene mapping to deconstruct the DFTD karyotype and determine the chromosome and gene rearrangements involved in carcinogenesis. Chromosome painting on three different DFTD tumour strains determined the origins of marker chromosomes and provided a general overview of the rearrangement in DFTD karyotypes. Mapping of 105 BAC clones by fluorescence in situ hybridisation provided a finer level of resolution of genome rearrangements in DFTD strains. Our findings demonstrate that only limited regions of the genome, mainly chromosomes 1 and X, are rearranged in DFTD. Regions rearranged in DFTD are also highly rearranged between different marsupials. Differences between strains are limited, reflecting the unusually stable nature of DFTD. Finally, our detailed maps of both the devil and tumour karyotypes provide a physical framework for future genomic investigations into DFTD