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

Formation Channels for Blue Straggler Stars

In this chapter we consider two formation channels for blue straggler stars: 1) the merger of two single stars via a collision, and 2) those produced via mass transfer within a binary. We review how computer simulations show that stellar collisions are likely to lead to relatively little mass loss and are thus effective in producing a young population of more-massive stars. The number of blue straggler stars produced by collisions will tend to increase with cluster mass. We review how the current population of blue straggler stars produced from primordial binaries decreases with increasing cluster mass. This is because exchange encounters with third, single stars in the most massive clusters tend to reduce the fraction of binaries containing a primary close to the current turn-off mass. Rather, their primaries tend to be somewhat more massive and have evolved off the main sequence, filling their Roche lobes in the past, often converting their secondaries into blue straggler stars (but more than 1 Gyr or so ago and thus they are no longer visible today as blue straggler stars).Comment: Chapter 9, in Ecology of Blue Straggler Stars, H.M.J. Boffin, G. Carraro & G. Beccari (Eds), Astrophysics and Space Science Library, Springe

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

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

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