Approximate lower bounds of the Weinstein and Temple variety

Abstract

By using the Weinstein interval or coupling the Temple lower bound to a variational upper bound one can in principle construct an error bar about the groundstate energy of an electronic system. Unfortunately there are theoretical and calculational issues which complicate this endeavor so that at best only an upper bound to the electronic energy has been practical in systems with more than a few electrons. The calculational issue is the complexity of ͗H 2 ͘ which is necessary in the Temple or Weinstein approach. In this work we provide a way to approximate the ͗H 2 ͘ to any desired accuracy using much simpler ͗H͘-like information so that the lower bound calculations are more practical. The helium atom is used as a testing ground in which we obtain approximate error bars for the ground-

Methods of Bilateral Approximations for Nonlinear Eigenvalue Problems

“…As a rule, it is impossible to apply (generalize) those methods that exist for linear problems to find the upper and lower bounds of eigenvalues of nonlinear spectral problems. Namely: various variants of the method of intermediate problems (Weinstein's method) (see, for example [6,7,8,9,10], as well as a bibliography in Gould [10], [11]), the Fichera method [12], as well as methods and algorithms based on inclusion theorems (see, for example, G. Temple [13], L. Collatz [14], and N. J. Lehmann [15,16], H. Behnke [17], M. G. Marmorino [11]). Therefore, the concept and apparatus of interval analysis are used to construct methods of bilateral approximations (see, for example [18,19]).…”

One Approach to Construction of Bilateral Approximations Methods for Solution of Nonlinear Eigenvalue Problems

“…Later on the method has been refined by N. Bazly, D. W. Fox, C. Beattie, and others (see [12][13][14] and the bibliography in [10,14,15] The method, different from the method of intermediate problems was proposed by G. Fichera [16]. This method does not require constructing a base operator with a known spectrum, but has a narrower scope of application.…”