Non-Weyl Resonance Asymptotics for Quantum Graphs

We consider the resonances of a quantum graph $\mathcal G$ that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of $\mathcal G$ in a disc of a large radius. We call $\mathcal G$ a \emph{Weyl graph} if the coefficient in front of this leading term coincides with the volume of the compact part of $\mathcal G$. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.Comment: 29 pages, 2 figure

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

Towards a theory of singular thought about abstract mathematical objects

This essay uses a mental files theory of singular thought—a theory saying that singular thought about and reference to a particular object requires possession of a mental store of information taken to be about that object—to explain how we could have such thoughts about abstract mathematical objects. After showing why we should want an explanation of this I argue that none of three main contemporary mental files theories of singular thought—acquaintance theory, semantic instrumentalism, and semantic cognitivism—can give it. I argue for two claims intended to advance our understanding of singular thought about mathematical abstracta. First, that the conditions for possession of a file for an abstract mathematical object are the same as the conditions for possessing a file for an object perceived in the past—namely, that the agent retains information about the object. Thus insofar as we are able to have memory-based files for objects perceived in the past, we ought to be able to have files for abstract mathematical objects too. Second, at least one recently articulated condition on a file’s being a device for singular thought—that it be capable of surviving a certain kind of change in the information it contains—can be satisfied by files for abstract mathematical objects

Decomposing the Essential Spectrum

We use C*-algebra theory to provide a new method of decomposing the eseential spectra of self-adjoint and non-self-adjoint Schrodinger operators in one or more space dimensions

Spectral Properties of Random Non-self-adjoint Matrices and Operators

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the non-self-adjoint Anderson model changes suddenly as one passes to the infinite volume limit.Comment: keywords: eigenvalues, spectral instability, matrices, computability, pseudospectrum, Schroedinger operator, Anderson mode

Semi-classical Analysis and Pseudospectra

We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudospectra and boundary conditions in the semi-classical limit