Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Measurement of the duration of X-ray lasing pumped by an optical laser pulse of picosecond duration

Measurements of the duration of X-ray lasing pumped with picosecond pulses from the VULCAN optical laser are obtained using a streak camera with 700 fs temporal resolution. Combined with a temporal smearing due to the spectrometer employed, we have measured X-ray laser pulse durations for Ni-like silver at 13.9 nm with a total time resolution of 1.1 ps. For Ni-like silver, the X-ray laser output has a steep rise followed by an approximately exponential temporal decay with measured full-width at half-maximum (FWHM) of 3.7 (+/-0.5) ps. For Ne-like nickel lasing at 23.1 nm, the measured duration of lasing is approximate to10.7 (+/-1) ps (FWHM). An estimate of the duration of the X-ray laser gain has been obtained by temporally resolving spectrally integrated continuum and resonance line emission. For Ni-like silver, this time of emission is approximate to22 (+/-2) ps (FWHM), while for Ne-like nickel we measure approximate to35 (+/-2) ps (FWHM). Assuming that these times of emission correspond to the gain duration, we show that a simple model consistently relates the gain durations to the measured durations of X-ray lasing. (C) 2002 Elsevier Science B.V. All rights reserved.