Minimax principle

The variational Dirac-Coulomb and the corresponding Levy-Leblond problems, in which the large and the small components are treated independently, are analyzed. Close similarities between these two variational problems are emphasized. Several examples in which the so called strong minimax principle is violated are discussed. 

An application of the variational principle to an unbounded from below Dirac-Coulomb eigenvalue problem, requires imposing upon the trial function certain conditions. Among these the most important are the symmetry properties, the asymptotic behaviour and the relations between the large and the small components of the wavefunction related to the so called kinetic balance [1,2,3]. In practical calculations an exact fulfilment of these conditions may be difficult or even impossible. Therefore a number of minimax principles [4-7] have been formulated in order to allow for some less restricted choice of the trial functions. There exist in the literature many either purely intuitive or derived from computational experience, rules which are commonly used as a guidance in generating basis sets for variational relativistic calculations. 

Fig. 3. The solid lines illustrate the weak minimax principle. They describe the dependence ofEon a when p = pmax ot (inDl andLl) and the dependence of E on L when S - Su]ax(T) (in D2 and L2). The broken lines and the broken lines with dots show the dependence of the energy hypersutface on a single parameter...

The variational procedure in a many-electron space may be considered as several consecutively executed variational procedures in one-electron spaces (the procedure may be iterative if a self-consistency is required). This means that fulfilment of the one-electron minimax principle is a necessary (but, in general, not sufficient) condition for the fulfilment of a similar principle in a many-electron case. Therefore one should not expect a many-electron generalization of Eq. (7) being valid when, say, parameters L and S are varied. 

We assume for simplicity that the two-electron atom is described by a Hamiltonian (29) in which //(I) and //(2) are the hydrogen-like Dirac Hamiltonians and h, 2) = 0. Apparently, after this simplification the problem is trivial since it becomes separable to two one-electron problems. Nevertheless we present this example because it sheds some light upon formulations of the minimax principle in a many-electron case. 

Besides, condition (5) is necessary for S to be an upper bound to the corresponding eigenvalue E. The question how to control the behavior of the variational energy by using rather weakly constrained variational trial functions, motivated the formulation of a number of minimax principles [4-6]. A detailed discussion and classification of these approaches has been given in ref. [7]. In most general terms, they are based on the following condition ... 

Nevertheless, this is impossible for Dirichlet boundaries. Really, the energy functional, Equation (2.1), has the same form both for the free and the Dirichlet problem and its value for a function with support in 2 (A) estimates the functional value for a free problem (one shall continue these functions to a whole space with zero values). Hence, the minimax principle (see [5], Sect. XIII.1) excludes the situation when there is any other limiting point for the sequence of energies E A) for A oo. The same is evident by the use of KatrieTs trick [64]. 

The minimax principle is pessimistic in the extreme. It assumes that, if any alternative is selected, the worst possible outcome will occur. The maximum cost associated with each alternative is examined, and the alternative that minimizes the maximum cost is selected. In general, the mathematiced formulation of the minimax principle is... 

If the minimax principle is adopted, is indicated because it results in minimum costs, assuming the worst possible conditions. 

The mirror image of the minimax principle, the maximin principle, may be applied when the matrix contains profits or revenue measures. In this case the most pessimistic view suggests that the alternative to select is the one that maximizes the minimum profit or revenue associated with each alternative. The mathematical formulation of the maximin principle is... 

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