Estimating bounds on the highest and lowest eigenvalues of any matrix

A method for estimating the bounds for the highest and lowest eigenvalues of a finite-dimensional matrix is presented. The method is tested for the Hamiltonian matrix for several particle-in-a-box-like systems. We also provide results for the well-studied benchmark case of the ro-vibrational states of H₃ ⁺, and consider bounds obtained for a completely random non symmetric matrix. Finally, we discuss how the error in a Chebychev expansion solution of quantum scattering depends on the error made in estimating the highest eigenvalue of the Hamiltonian matrix. Received: 7 May 1999 / Accepted: 30 July 1999 / Published online: 15 December 1999     

Trades and Graphs

The trade spectrum of a simple graph G is defined to be the set of all t for which it is possible to assemble together t copies of G into a simple graph H, and then disassemble H into t entirely different copies of G. Trade spectra of graphs have applications to intersection problems, and defining sets, of G-designs. In this investigation, we give several constructions, both for specific families of graphs, and for graphs in general. Received: October 3, 1997 Final version received: May 7, 1999