No abstract

Abstract. 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 L q (0, 1) whenever 1 < q < ∞.

This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrödinger 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.

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 analyzed. 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.

No abstract

We study the quantization of the regularized hamiltonian, H, of the compactified D = 11 supermembrane with nontrivial 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.Date: 17 th February 2005.

Abstract. 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 finitedimensional subspace L ⊂ 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.

We establish sufficiency conditions in order to achieve approximation to discrete eigenvalues of self-adjoint operators in the second-order projection method suggested recently by Levitin and Shargorodsky, [15]. We find explicit estimates for the eigenvalue error and study in detail two concrete model examples. Our results show that, unlike the majority of the standard methods, second-order projection strategies combine non-pollution and approximation at a very high level of generality.