Let H0=−Δ+V0false(xfalse) be a Schrödinger operator on L2false(Rνfalse), ν=1,2, or 3, where V0false(xfalse) is a bounded measurable real‐valued function on Rν. Let V be an operator of multiplication by a bounded integrable real‐valued function V(x) and put Hr=H0+rV for real r. We show that the associated spectral shift function (SSF) ξ admits a natural decomposition into the sum of absolutely continuous and singular SSFs. In particular, the singular SSF is integer‐valued almost everywhere, even within the absolutely continuous spectrum where the same cannot be said of the SSF itself. This is a special case of an analogous result for resolvent comparable pairs of self‐adjoint operators, which generalises the case of a trace class perturbation appearing in [2] while also simplifying its proof. We present two proofs which demonstrate the equality of the singular SSF with two a priori different and intrinsically integer‐valued functions which can be associated with the pair H0, V: the total resonance index [3] and the singular μ‐invariant [2].
Articles you may be interested inNonoptical excited state spectroscopy of CF3Cl, CF2Cl2, and CFCl3: Bethe surfaces, and absolute transition probability measurement of preionizationedge valence and Rydberg transitions by angleresolved electron energy loss spectroscopy Nonoptical excited state spectroscopy of CHF2Cl: Characterization of nondipole n→σ* valence transitions by angleresolved electron energy loss spectroscopy A coplanar electron spectrometer has been constructed for angle-resolved electron energy loss measurements. . The experimental arrangement and performance of the spectrometer are described. Absolute absorption transition probabilities (or generalized oscillator strengths) of valence-shell (7-70 eV) electronic transitions of SF 6 have been determined as a function of energy loss and momentum transfer at an impact energy of 2.5 keY. New nondipole transitions have been observed at nonzero momentum transfer. Together with the term values obtained from previous dipole electron energy loss and photoabsorption measurements, the momentum' transfer dependence of the transition probabilities can be used to provide tentative assignments of the observed nondipole valence-shell transitions of SF 6. Despite the complexity of the electronic structure of SF 6 , the present work demonstrates the feasibility of angle-resolved electron energy loss spectroscopy for investigating nondipole phenomena and related electron-induced excitation processes.3390
In this paper we prove for rank one perturbations that the imaginary part of a resonance point is inversely proportional by a factor of −2 to the rate of change of the scattering phase, as a function of the coupling parameter, evaluated at the real part of the resonance point. This equality is in agreement with the Breit-Wigner formula from quantum scattering theory. For more general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of resonance points, non-real and real.2010 Mathematics Subject Classification. Primary 47A40, 47A55, 47A70.
Given a self-adjoint operator and a relatively trace class perturbation, one can associate the singular spectral shift function – an integer-valued function on the real line which measures the flow of singular spectrum, not only at points outside of the essential spectrum, where it coincides with the classical notion of spectral flow, but at points within the essential spectrum too.
The singular spectral shift function coincides with both the total resonance index and the singular μ-invariant.
In this paper we give a direct proof of the equality of the total resonance index and singular μ-invariant assuming only the limiting absorption principle and no condition of trace class type – a context in which the existence of the singular spectral shift function is an open question.
The proof is based on an application of the argument principle to the poles and zeros of the analytic continuation of the scattering matrix considered as a function of the coupling parameter.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.