Lyonell Boulton

On approximation of the eigenvalues of perturbed periodic Schrödinger operators

This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so-called quadratic projection method, in order to achieve convergence free from spectral pollution. We describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples.

Discreteness of the spectrum of the compactified D=11 supermembrane with nontrivial winding

Abstract We analyze the Hamiltonian of the compactified D=11 supermembrane with nontrivial central charge in terms of the matrix model constructed in [Phys. Rev. D 66 (2002) 045023, hep-th/0103261 ]. Our main result provides a rigorous proof that the quantum Hamiltonian of the supersymmetric model has compact resolvent and thus its spectrum consists of a discrete set of eigenvalues with finite multiplicity.

On the spectrum of a matrix model for the D = 11 supermembrane compactified on a torus with non-trivial winding

The spectrum of the Hamiltonian of the double compactified D = 11 supermembrane with non-trivial central charge or, equivalently, the non-commutative symplectic super Maxwell theory, is analysed. In distinction to what occurs for the D = 11 supermembrane in Minkowski target space where the bosonic potential presents string-like spikes which render the spectrum of the supersymmetric model continuous, we prove that the potential of the bosonic compactified membrane with non-trivial central charge is strictly positive definite and becomes infinity in all directions when the norm of the configuration space goes to infinity. This ensures that the resolvent of the bosonic Hamiltonian is compact. We find an upper bound for the asymptotic distribution of the eigenvalues.

Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials

We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie on the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-side estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method as well as illustrate our results with various numerical experiments.

Basis properties of eigenfunctions of the p-laplacian

For p ≥ 12 11 the eigenfunctions of the non-linear eigenvalue problem for the p-Laplacian on the interval (0,1) are shown to form a Riesz basis of L 2 (0,1) and a Schauder basis of Lq(0,1) whenever 1 < q < oo.

The heat kernel of the compactified D=11 supermembrane with non-trivial winding

We study the quantization of the regularized Hamiltonian, H, of the compactified D=11 supermembrane with non-trivial winding. By showing that H is a relatively small perturbation of the bosonic Hamiltonian, we construct a Dyson series for the heat kernel of H and prove its convergence in the topology of the von Neumann–Schatten classes so that e−Ht is ensured to be of finite trace. The results provided have a natural interpretation in terms of the quantum mechanical model associated to regularizations of compactified supermembranes. In this direction, we discuss the validity of the Feynman path integral description of the heat kernel for D=11 supermembranes and obtain rigorously a matrix Feynman–Kac formula.

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 convergence of second-order spectra and multiplicity

The notion of second-order relative spectrum of a self-adjoint operator acting on a Hilbert space has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. In this paper we examine how the second-order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of the underlying operator. Our theoretical findings are supported by various numerical experiments on the computation of guaranteed eigenvalue inclusions via finite element bases.

Limiting set of second order spectra

Let M be a self-adjoint operator acting on a Hilbert space H. A complex number z is in the second order spectrum of M relative to a finite-dimensional subspace L C Dom M 2 iff the truncation to L of (M - z) 2 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 M, the uniform limit of these sets, as L increases towards H, contain the isolated eigenvalues of M 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.

The supermembrane with central charges: (2 + 1) D NCSYM, confinement and phase transition

The spectrum of the bosonic sector of the D=11 supermembrane with central charges is shown to be discrete and with finite multiplicities, hence containing a mass gap. The result extends to the exact theory our previous proof of the similar property for the SU(N) regularised model and strongly suggest discreteness of the spectrum for the complete Hamiltonian of the supermembrane with central charges. This theory is a quantum equivalent to a symplectic non-commutative super-Yang–Mills in 2+1 dimensions, where the space-like sector is a Riemann surface of positive genus. In this context, it is argued how the theory in 4D exhibits confinement in the N=1 supermembrane with central charges phase and how the theory enters in the quark–gluon plasma phase through the spontaneous breaking of the centre. This phase is interpreted in terms of the compactified supermembrane without central charges.