arXiv subjects

Lyonell Boulton

Publications and source records attributed to Lyonell Boulton.

39 records · Page 3Linked to original sources

arXiv2005

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 a matrix Feynman-Kac formula.

hep-th↗

arXiv2004

Finite lifetime eigenfunctions of coupled systems of harmonic oscillators

We find a Hermite-type basis for which the eigenvalue problem associated to the operator $H_{A,B}:=B(-\partial_x^2)+Ax^2$ acting on $L^2({\bf R};{\bf C}^2)$ becomes a three-terms recurrence. Here $A$ and $B$ are two constant positive definite matrices with no other restriction. Our main result provides an explicit characterization of the eigenvectors of $H_{A,B}$ that lie in the span of the first four elements of this basis when $AB\not= BA$.

math.SP↗

arXiv2003

Limiting set of second order spectra

M.Levitin and E.Shargorodsky purposed in a recent article, [math.SP/0212087], the use of the so called ``second order relative spectrum'', to find eigenvalues of self-adjoint operators in gaps of the essential spectrum. Let $M$ be a self-adjoint operator acting on the Hilbert space $H$. A complex number $z$ is in the second order spectrum of $M$ relative to a finite dimensional subspace $Λ\subset dom M^2$ if and only if the truncation to $Λ$ of $(M-z)^2$ is not invertible. It is remarkable that these sets seem to provide a general method to estimating eigenvalues free from the problems of spectral pollution present in most linear methods. In this notes we investigate rigorously various aspects of the use of second order spectrum to finding eigenvalues. Our main result shows that, under certain fairly mild hypothesis on $M$, the uniform limit of the second order spectra, as $Λ$ increases toward $H$, contains the isolated eigenvalues of $M$ of finite multiplicity. In applications the essential spectrum can be computed analytically, while precisely these eigenvalues are the ones that should be approximated numerically. Hence this method seems to combine non-pollution and approximation at a very high level of generality.

math.SP↗