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Moment problem formulation of the Schrödinger equation

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Abstract

We show that it is possible to project out in an exact manner the lowest eigenstate of Schrödinger equations. Taking into account the nodeless property of the lowest eigenstate one can replace the full Schrödinger equation by a moment problem whose measure is the eigenstate itself. The infinite set of positivity inequalities linked to this moment problem provides a framework which allows to compute sequences of upper and lower bounds to the unknown eigenvalue and eigenfunction.

The effective computation is based on deep convexity properties embedded in the set of hierarchical inequalities associated to this moment problem. The convexity allows to get the bounds through linear programming. We illustrate the method with simple one dimensional problems.

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Authors and Affiliations

  1. Service de Physique Théorique, CE-Saclay, F-91191, Gif-sur-Yvette Cedex, France

    Daniel Bessis

  2. Center for Theoretical Study of Physical Systems, Clark-Atlanta University, 30314, Atlanta, GA, USA

    Daniel Bessis

  3. Center for Theoretical Study of Physical Systems, Clark-Atlanta University, 30314, Atlanta, GA, USA

    Carlos R. Handy

Authors
  1. Daniel Bessis
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  2. Carlos R. Handy
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Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique.

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Bessis, D., Handy, C.R. Moment problem formulation of the Schrödinger equation. Numer Algor 3, 1–16 (1992). https://doi.org/10.1007/BF02141911

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  • DOI: https://doi.org/10.1007/BF02141911

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