Spectral properties in supersymmetric matrix models

We formulate a general sufficiency criterion for discreteness of the spectrum of both supersymmmetric and non-supersymmetric theories with a fermionic contribution. This criterion allows an analysis of Hamiltonians in complete form rather than just their semiclassical limits. In such a framework we examine spectral properties of various (1+0) matrix models. We consider the BMN model of M-theory compactified on a maximally supersymmetric pp-wave background, different regularizations of the supermembrane with central charges and a non-supersymmetric model comprising a bound state of N D2 with m D0. While the first two examples have a purely discrete spectrum, the latter has a continuous spectrum with a lower end given in terms of the monopole charge.     

Basis properties of eigenfunctions of the -Laplacian

For , the eigenfunctions of the non-linear eigenvalue problem for the -Laplacian on the interval are shown to form a Riesz basis of and a Schauder basis of whenever .

     

On pseudospectra of matrix polynomials and their boundaries

In the first part of this paper (Sections 2-4), the main concern is with the boundary of the pseudospectrum of a matrix polynomial and, particularly, with smoothness properties of the boundary. In the second part (Sections 5-6), results are obtained concerning the number of connected components of pseudospectra, as well as results concerning matrix polynomials with multiple eigenvalues, or the proximity to such polynomials.

     

On the Stability of a Forward–Backward Heat Equation

In this paper we examine spectral properties of a family of periodic singular Sturm?CLiouville problems which are highly non-self-adjoint but have purely real spectrum. The problem originated from the study of the lubrication approximation of a viscous fluid film in the inner surface of a rotating cylinder and has received a substantial amount of attention in recent years. Our main focus will be the determination of Schatten class inclusions for the resolvent operator and regularity properties of the associated evolution equation.     

Limiting set of second order spectra

Let be a self-adjoint operator acting on a Hilbert space . A complex number is in the second order spectrum of relative to a finite-dimensional subspace iff the truncation to of is not invertible. This definition was first introduced in Davies, 1998, and according to the results of Levin and Shargorodsky in 2004, these sets provide a method for estimating eigenvalues free from the problems of spectral pollution. In this paper we investigate various aspects related to the issue of approximation using second order spectra. Our main result shows that under fairly mild hypothesis on the uniform limit of these sets, as increases towards , contain the isolated eigenvalues of of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine nonpollution and approximation at a very high level of generality.

     

Projection Methods for Discrete Schrodinger Operators

Let H be the discrete Schrödinger operator acting on l² Z⁺, where the potential v is real-valued and v(n) 0 as n . Let P be the orthogonal projection onto a closedlinear subspace l² Z⁺). In a recent paper E. B. Davies definesthe second order spectrum Spec₂(H, ) of H relative to as theset of z C such that the restriction to of the operator P(H- z)²P is not invertible within the space . The purpose of thisarticle is to investigate properties of Spec₂(H, ) when islarge but finite dimensional. We explore in particular the connectionbetween this set and the spectrum of H. Our main result providessharp bounds in terms of the potential v for the asymptoticbehaviour of Spec₂(H, ) as increases towards l² Z⁺). 2000 MathematicsSubject Classification 47B36 (primary), 47B39, 81-08 (secondary).     

A Prüfer angle approach to singular Sturm–Liouville problems with Molčanov potentials

Singular Sturm–Liouville problems for -y^″+qy=λy$-y^{″}+\mathit{qy}=\lambda y$ on (0,∞)$(0,\infty )$ are studied for potentials q which are bounded below and satisfy Mol?anov's necessary and sufficient condition for discrete spectrum. A Prüfer angle approach is given for eigenvalue location and eigenfunction oscillation, paralleling that for the regular case. In particular, the eigenvalues are characterized by a “right-hand boundary condition” even though q is of limit point type.     

Non-variational approximation of discrete eigenvalues of self-adjoint operators

^** Email: L.Boulton{at}ma.hw.ac.uk We establish sufficient conditions for approximation of discreteeigenvalues of self-adjoint operators in the second-order projectionmethod suggested recently in Levitin & Shargorodsky (2004,Spectral pollution and second order relative spectra for self-adjointoperators. IMA J. Numer. Anal., 24, 393–416). We findfairly explicit estimates for the eigenvalue error and studyin detail two concrete model examples. Our results show thatsecond-order projection strategies not only are universallypollution free but also achieve approximation under naturalconditions on the discretising basis.     

On a class of non-self-adjoint periodic eigenproblems with boundary and interior singularities

We prove that all the eigenvalues of a certain highly non-self-adjoint Sturm-Liouville differential operator are real. The results presented are motivated by and extend those recently found by various authors (Benilov et al. (2003) [3], Davies (2007) [7] and Weir (2008) [18]) on the stability of a model describing small oscillations of a thin layer of fluid inside a rotating cylinder.