Generating quantum energy bounds by the moment method: A linear-programming approach

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Abstract
Recently, Handy and Bessis derived, for singular multidimensional Schrödinger equations, infinite sets of hierarchical inequalities fulfilled by the moments of the ground-state wave function. These inequalities, when increasing in number, provide tighter lower and upper bounds to the ground-state energy. This analysis makes use of nonlinear multivariate determinant functionals (Hankel-Hadamard determinants) which define complicated convex sets in the space of the ‘‘missing moments.’’ It is possible to reformulate all the above in terms of linear programming. The previous sets are now replaced by new sets defined by linear inequalities. This new formulation makes the general moment approach highly practical. We describe the general formalism and give a simple one-dimensional example to illustrate it. The three-dimensional quadratic Zeeman problem, in the transition region, is analyzed by this method. Preliminary results for values of the magnetic field B=0.2, 2, and 20 (in atomic units) are given.