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

Handy, Carlos R.; Bessis, D.; Morley, T. D.

doi:10.1103/physreva.37.4557

Cited by 50 publications

(37 citation statements)

References 13 publications

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“…The most famous of these was the notoriously difficult, strong coupling, singular perturbation problem corresponding to determining the ground state binding energy of the quadratic Zeeman effect for super-strong magnetic fields (of the type encountered in neutron stars). Handy et al [24,25] were able to extend the EMM analysis to such problems confirming, through tight bounds, the results of Le Guillou and Zinn-Justin [26] derived through more sophisticated, analytical methods.…”

Section: Preliminariessupporting

confidence: 57%

“…An alternative approach (for multidimensional systems) is to use LUdecomposition methods on the (multidimensional) Hankel moment matrix (for the weight), in order to generate the same orthogonal polynomials [39]. One interesting application of this is to use EMM [20,24,25] to determine the power moments of the physical (positive) bosonic ground state (through tight bounds), for a given system. These can then be incorporated within OPPQ to generate the discrete state energies for all of the desired discrete states.…”

Section: Oppq and Hd: General Quantization Strategies Motivated By Exmentioning

confidence: 99%

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Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure

et al. 2015

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Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrödinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states (generated through the power moments) can define an L 2 convergent, non-orthonormal, basis expansion with sufficient pointwise convergent behaviors, enabling the direct coupling of the global (power moments) and local (Taylor series) expansions in configuration space. Our formulation is elaborated within the orthogonal polynomial projection quantization (OPPQ) configuration space representation previously developed The quantization approach pursued here defines an alternative strategy emphasizing the relevance of OPPQ to the reconstruction of the local structure of the physical states.

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Section: Preliminariessupporting

confidence: 57%

Section: Oppq and Hd: General Quantization Strategies Motivated By Exmentioning

confidence: 99%

Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure

et al. 2015

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“…see Curto and Fialkow [43] or Handy et al [46]. Finally, for ease of notation in Section 2, we define:…”

Section: Conflicts Of Interestmentioning

confidence: 99%

A Comparative Study of B-, Γ- and Log-Normal Distributions in a Three-Moment Parameterization for Drop Sedimentation

Ziemer

Wacker

2014

Atmosphere

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This paper examines different distribution functions used in a three-moment parameterization scheme with regard to their influence on the implementation and the results of the parameterization scheme. In parameterizations with moment methods, the prognostic variables are interpreted as statistical moments of a drop size distribution, for which a functional form has to be assumed. In cloud microphysics, parameterizations are frequently based on gamma-and log-normal distributions, while for particle-laden flows in engineering, the beta-distribution is sometimes used. In this study, the three-moment schemes with beta-, gamma-and log-normal distributions are tested in a 1D framework for drop sedimentation, and their results are compared with those of a spectral reference model. The gamma-distribution performs best. The results of the parameterization with the beta-and the log-normal distribution have less similarity to the reference solution, particularly with regard to number density and rain rate. Theoretical considerations reveal that (depending on the type of the distribution function) only selected combinations of moments can be predicted together. Among these is the important combination of "number density, liquid water content, radar reflectivity" for all three distributions. Advection or source/sink terms can only be calculated under certain conditions on the moment values (positivity of the Hankel-Hadamard determinant). These are derived from mathematical theory ("moment problem") and are more restrictive for three-moment than for two-moment schemes.Atmosphere 2014, 5 485

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“…This revealed the anomalous kink behavior of the ν-moment equation for the QES states; however the focus of that first work was on bounding the non-QES energies for the ground and first excited states. In subsequent works [15,16] Handy was able to transform the nonlinear version of EMM into an equivalent linear programming based formulation which allowed for its implementation to a broad range of multidimensional systems, including the notoriously difficult quadratic Zeeman effect for superstrong magnetic fields [15,16]. The relevance of moment representations for quantizing singular perturbation-strongly coupled systems had been previously noted by Handy [22], in the context of finding a more rigorous alternative to lattice high temperature expansions in field theory.…”

Section: The Bessis Representation: Relevance Of the Eigenvalue Momenmentioning

confidence: 99%

“…The existence of QES states as due to a nonuniform moment order structure was known to Handy and Bessis (HB) in the context of their development of the Eigenvalue Moment Method (EMM) [14][15][16], the first application of semidefinite programming (SDP) [17][18] in quantum physics. We ouline the relevant history since it impacts this work, and underscores the importance of moment representations for quantizing physical systems.…”

Section: The Bessis Representation: Relevance Of the Eigenvalue Momenmentioning

confidence: 99%

A moments' analysis of quasi-exactly solvable systems: a new perspective on the sextic potential $gx^6+bx^4+mx^2 +{\beta \over {x^2}}$

Handy¹,

Vrînceanu²,

Gupta³

2014

J. Phys. A: Math. Theor.

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

Cited by 50 publications

References 13 publications

Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure

Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure

A Comparative Study of B-, Γ- and Log-Normal Distributions in a Three-Moment Parameterization for Drop Sedimentation

A moments' analysis of quasi-exactly solvable systems: a new perspective on the sextic potential $gx^6+bx^4+mx^2 +{\beta \over {x^2}}$

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