On QZ steps with perfect shifts and computing the index of a differential-algebraic equation

In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

From Quantum Field Theory to the Contemporary Quantum Mechanics

In this talk we remind how the notion of the so-called clothed particles, put forward in relativistic quantum field theory by Greenberg and Schweber, can be used via the method of unitary clothing transformations (shortly, the UCT method) when finding the eigenstates of the total Hamiltonian H in case of interacting fields with the Yukawa - type couplings. In general, the UCT method is aimed at reduction of the exact eigenvalue problem in the primary Fock space to the model-space problems in the corresponding Hilbert spaces of the contemporary quantum mechanics. In this context we consider an approximate treatment of the physical vacuum, the observable one-particle and two-particle bound and scattering states.

Forced Harmonic Response of a Continuous System Displaying Eigenvalue Veering Phenomena

Prior studies of self-adjoint linear vibratory systems have extensively explored the free vibration phenomena associated with veering of eigenvalue loci that depict the dependence of natural frequencies on a system parameter. The present work is an exploration of the effect of such phenomena on the response of a nearly-periodic continuum to harmonic excitation. The focus of the analysis is the prototypical system of a two-span beam with a strong torsional spring at the intermediate pin support. The results of an exact eigenvalue analysis, not previously disclosed, are used to perform a modal analysis of the steady-state response to a harmonic concentrated force applied to the middle of one span. The modal equations are used to identify situations in which the force response is localized to one span, as well as the degree to which the location and magnitude of the peak displacement display parameter sensitivity. The effect of hysteretic loss on forced localization is discussed.