Searcharxiv⌕ Search

arXiv subjects

Rafael Tiedra de Aldecoa

Publications and source records attributed to Rafael Tiedra de Aldecoa.

27 records · Page 2Linked to original sources
arXiv2008

Time delay for dispersive systems in quantum scattering theory

We consider time delay and symmetrised time delay (defined in terms of sojourn times) for quantum scattering pairs $\{H_0=h(P),H\}$, where $h(P)$ a dispersive operator of hypoelliptic-type. For instance $h(P)$ can be one of the usual elliptic operators such as the Schrödinger operator $h(P)=P^2$ or the square-root Klein-Gordon operator $h(P)=\sqrt{1+P^2}$. We show under general conditions that the symmetrised time delay exists for all smooth even localization functions. It is equal to the Eisenbud-Wigner time delay plus a contribution due to the non-radial component of the localization function. If the scattering operator $S$ commutes with some function of the velocity operator $\nabla h(P)$, then the time delay also exists and is equal to the symmetrised time delay. As an illustration of our results we consider the case of a one-dimensionnal Friedrichs Hamiltonian perturbed by a finite rank potential. Our study put into evidence an integral formula relating the operator of differentiation with respect to the kinetic energy $h(P)$ to the time evolution of localization operators.

math-ph↗
arXiv2007

Anisotropic Lavine's formula and symmetrised time delay in scattering theory

We consider, in quantum scattering theory, symmetrised time delay defined in terms of sojourn times in arbitrary spatial regions symmetric with respect to the origin. For potentials decaying more rapidly than $|x|^{-4}$ at infinity, we show the existence of symmetrised time delay, and prove that it satisfies an anisotropic version of Lavine's formula. The importance of an anisotropic dilations-type operator is revealed in our study.

math-ph↗
arXiv2007

Spectral analysis for convolution operators on locally compact groups

We consider operators $H_μ$ of convolution with measures $μ$ on locally compact groups. We characterize the spectrum of $H_μ$ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of $H_μ$, respectively. The proofs rely on commutator methods.

math-ph↗
arXiv2006

On the spectrum of magnetic Dirac operators with Coulomb-type perturbations

We carry out the spectral analysis of singular matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the perturbations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Constant, periodic as well as diverging magnetic fields are covered, and Coulomb potentials up to the physical nuclear charge Z<137 are allowed. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.

math-ph↗
arXiv2005

Time delay and short-range scattering in quantum waveguides

Although many physical arguments account for using a modified definition of time delay in multichannel-type scattering processes, one can hardly find rigorous results on that issue in the literature. We try to fill in this gap by showing, both in an abstract setting and in a short-range case, the identity of the modified time delay and the Eisenbud-Wigner time delay in waveguides. In the short-range case we also obtain limiting absorption principles, state spectral properties of the total Hamiltonian, prove the existence of the wave operators and show an explicit formula for the S-matrix. The proofs rely on stationary and commutator methods.

math-ph↗
arXiv2004

The nature of the essential spectrum in curved quantum waveguides

We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou and Dermenjian et al. As a technical preliminary, we carry out a spectral analysis for Schrodinger-type operators in straight Dirichlet tubes. We also apply the result to the strips embedded in abstract surfaces.

math-ph↗
arXiv2004

On perturbations of Dirac operators with variable magnetic field of constant direction

We carry out the spectral analysis of matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the pertubations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Various situations, for example when the magnetic field is constant, periodic or diverging at infinity, are covered. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.

math-ph↗