J. P. da Providência

Computing the numerical range of Krein space operators

Abstract Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(T) in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined elliptical and hyperbolical numerical ranges. The numerical results reported here indicate that this method is very efficient, since it is faster and more accurate than either of the existing algorithms. Further, it may yield easy solutions for the inverse indefinite numerical range problem. Our algorithm uses an idea of Marcus and Pesce from 1987 for generating Hilbert space numerical ranges of matrices of size n.

Tsallis entropies and matrix trace inequalities in quantum statistical mechanics

Extensions of trace inequalities arising in statistical mechanics are derived in a straightforward and unified way from a variational characterization of the Tsallis entropy. Namely, one-parameter extension of the thermodynamic inequality is presented and its equivalence to a generalized Peierls–Bogoliubov inequality is stated, in the sense that one may be obtained from the other, and vice versa.

Collective modes in metals: quantal and semiclassical approaches

A quantal description (da Provid?ncia Jr and Barber?n 1992 Phys. Rev. B?45 6935?7; Barber?n and da Provid?ncia Jr 1996 Z. Phys. B 101 3?12) of the collective modes in metals is revisited. Collective variables are explicitly introduced. A time-dependent Slater determinant is assumed for the wavefunction of the system of valence electrons. Analogous models are also investigated using the semiclassical approximation based on the Wigner transform. The results indicate that for those collective modes, which are associated with a large number of particles, namely the cloud of valence electrons, the semiclassical approximation provides good results and is a reliable tool.

Non-Hermitian Quantum Mechanics of Bosonic Operators

The spectral analysis of a family of non-Hermitian operators appearing in quantum physics is our main concern. The properties of such operators are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operators and their adjoints can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.

Generalized Rayleigh quotients and generating vectors

This article is concerned with the following problem: given , of which either A or B is Hermitian, and given , determine a vector such that with , referred to as a generalized Rayleigh quotient. Such a vector w is called a generating vector for z. Our approach builds on the fact that the field of values of a linear pencil can be accurately approximated under the compression into bidimensional linear pencils, in which case the problem has an exact solution. The efficiency of the presented algorithms is discussed and numerical examples are included.