Upper and lower bounds for an eigenvalue associated with a positive eigenvector

Mouchet, Amaury

doi:10.1063/1.2168124

Cited by 4 publications

(8 citation statements)

References 30 publications

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“…We did not find this problem listed in the more or less classical geometrical optimization problems posed in R 3 [1,51]. The following partial results have been obtained in ( [47], Sect. IV.B):…”

Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 69%

“…As interesting as the approximate resolution of (P N ) for some values of N , is the asymptotic behavior of the optimal value and solution sets of (P N ) when N → +∞. In that respect, the following general conjecture was posed in [47]:…”

Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 99%

“…The global maximum of F 4 , a function of 12 variables, is 6; it is attained when the four points form a regular tetrahedron. For larger N , numerical investigations led the author of [47] to the following three conjectures:…”

Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 99%

“…As for most of the problems of this kind, the following originates in Physics [46,47]. It was posed to us by Ley and Mouchet (university of Tours, France): Find the geometrical configuration formed by N particles so as to minimize a given energy function.…”

Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 99%

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A new series of conjectures and open questions in optimization and matrix analysis

Hiriart-Urruty¹

2008

ESAIM: COCV

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Abstract. We Introduction"Some problems open doors, some problems close doors, and some remain curiosities, but all sharpen our wits and act as a challenge and a test of our ingenuity and techniques". So said Atiyah in the preface to Mathematics: frontiers and perspectives (2000, by the International Mathematical Union, published by the American Mathematical Society). This statement is, in our opinion, a good introduction to what a collection of problems is or should be. In mathematics, each area or subarea produces its own lists of (more or less celebrated) problems and open questions, sometimes hard to appreciate or just to understand if one does not work in the concerned field, as given evidence by the lists of problems offered in the above-referenced book.Our objective here is more modest: we expose a selected list of questions that all belong to Optimization or Matrix analysis. They are of unequal importance and diverse origins, but all of rather wide interest. Some have a theoretical flavour, some others clearly hinge on calculation, and what it is asked for each of them varies. There are problems about which we know practically nothing, some others have partial answers; and also some are already solved but in rather indirect manners; for them we would like to have more natural, or at least different, proofs.For each of the nine problems described in this paper, we propose a short presentation, the state of the art and a list of appropriate references. As a result, each of the problems listed can be read independently of the others, according to the interest or knowledge of the reader. The reader could thus try to tackle some of them, that is at least our aim in writing down such a paper.Keywords and phrases. Convex sets, positive (semi)definite matrices, variational problems, energy functions, global optimization, permanent function, bistochastic matrices, normal matrices.

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Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 69%

Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 99%

Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 99%

Section: Problem Minimization Of An Energy For a General Coulombian Pmentioning

confidence: 99%

See 2 more Smart Citations

A new series of conjectures and open questions in optimization and matrix analysis

Hiriart-Urruty¹

2008

ESAIM: COCV

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“…Besides the variational and perturbative techniques that are mentioned above, there exists a third very general method for approximating the ground-state energy of a quantum system, namely the differential method (see [15,16, and references therein for a historical track]) whose starting point is recalled in section 2. for the sake of completeness. As for the variational methods, the differential method call on a family of trial functions that supposedly mimic the ground-state and that allow for the construction of a function (the average of the Hamiltonian in the former case, the so-called local energy in the latter case) whose absolute extrema within the chosen trial family provide bounds on the exact ground-state energy E 0 .…”

Section: Introductionmentioning

confidence: 99%

Bounding the ground-state energy of a many-body system with the differential method

Mouchet¹

2006

Nuclear Physics A

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This paper promotes the differential method as a new fruitful strategy for estimating a ground-state energy of a many-body system. The case of an arbitrary number of attractive Coulombian particles is specifically studied and we make some favorable comparison of the differential method to the existing approaches that rely on variational principles. A bird's-eye view of the treatment of more general interactions is also given.

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Potential envelope theory and the local energy theorem

Gibara

Hall

2019

Journal of Mathematical Physics

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We consider a one-particle bound quantum mechanical system governed by a Schrödinger operator H = −∆ + v f (r), where f (r) is an attractive central potential, and v > 0 is a coupling parameter. If φ ∈ D(H ) is a 'trial function', the local energy theorem tells us that the discrete energies of H are bounded by the extreme values of (H φ)/φ, as a function of r. We suppose that f (r) is a smooth transformation of the form f = g(h), where g is monotone increasing with definite convexity and h(r) is a potential for which the eigenvalues Hn(u) of the operator H = −∆ + u h(r), for appropriate u > 0, are known. It is shown that the eigenfunctions of H provide local-energy trial functions φ which necessarily lead to finite eigenvalue approximations that are either lower or upper bounds. This is used to extend the local energy theorem to the case of upper bounds for the excited-state energies when the trial function is chosen to be an eigenfunction of such an operator H. Moreover, we prove that the local-energy approximations obtained are identical to 'envelope bounds', which can be obtained directly from the spectral data Hn(u) without explicit reference to the trial wave functions.PACS numbers: 03.65.Ge.

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Upper and lower bounds for an eigenvalue associated with a positive eigenvector

Cited by 4 publications

References 30 publications

A new series of conjectures and open questions in optimization and matrix analysis

A new series of conjectures and open questions in optimization and matrix analysis

Bounding the ground-state energy of a many-body system with the differential method

Potential envelope theory and the local energy theorem

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