Sectorial perturbations of self-adjoint matrices and operators

This paper considers $N\times N$ matrices of the form $A_\gamma =A+ \gamma B$, where $A$ is self-adjoint, $\gamma \in C$ and $B$ is a non-self-adjoint perturbation of $A$. We obtain some monodromy-type results relating the spectral behaviour of such matrices in the two asymptotic regimes $|\gamma |\to\infty$ and $|\gamma |\to 0$ under certain assumptions on $B$. We also explain some properties of the spectrum of $A_\gamma$ for intermediate sized $\gamma$ by considering the limit $N\to\infty$, concentrating on properties that have no self-adjoint analogue. A substantial number of the results extend to operators on infinite-dimensional Hilbert spaces.Comment: 5 figure

Spectral Theory of Pseudo-Ergodic Operators

We define a class of pseudo-ergodic non-self-adjoint Schr\"odinger operators acting in spaces $l^2(X)$ and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson model acting on $l^2(\Z)$, and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators.Comment: 22 page

An Indefinite Convection-Diffusion Operator

We give a mathematically rigorous analysis which confirms the surprising results in a recent paper of Benilov, O'Brien and Sazonov about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics.Comment: Preprint, 13 page

Semi-classical States for Non-self-adjoint Schrodinger Operators

We prove that the spectrum of certain non-self-adjoint Schrodinger operators is unstable in the semi-classical limit. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semi-classical modes of the operator by the JWKB method for energies far from the spectrum

Triviality of the Peripheral Point Spectrum

If T_t=\rme^{Zt} is a positive one-parameter contraction semigroup acting on $l^p(X)$ where $X$ is a countable set and $1\leq p <\infty$, then the peripheral point spectrum $P$ of $Z$ cannot contain any non-zero elements. The same holds for Feller semigroups acting on $L^p(X)$ if $X$ is locally compact

Personal space : bring on the physics revolution

Some years ago a student submitted a practical assignment in which he wrote something along these lines: I collected the data on Sauchiehall Street on Friday afternoon. I asked any young-looking males (who didnt look too scary!) to fill in the questionnaire. It started to rain about four oclock so I went in Costa Coffee, and when I came out there werent so many people about, so I finished it off on Saturday morning. Colleagues felt this was inappropriate in a practical essay on a scientific subject. They objected to the use of the word I, which by definition made it a subjective account; and they suggested that a phrase such as Data were collected from a random sample of young males would have been more suitable. But I disagreed strongly, arguing that the student account was more informative, more scientific, more honest, and there was no attempt to hide behind scientific rhetoric. And obviously, the sample could not be called random