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Physical Review A

Moment-method analysis of the ground state of discretized bosonic systems

Carlos R. Handy, Jian Qun Pei

Published Oct 1, 1988• 2 citations
Publisher

Abstract

The eigenvalue moment method for bosonic systems, recently developed by C. R. Handy and D. Bessis [Phys. Rev. Lett 55, 931 (1985)], is extended to discrete quantum mechanics. The relevant formalism is described and applied in the context of the discretized harmonic and sextic anharmonic potentials. Rapidly converging bounds to the associated ground-state energies are obtained for fixed lattice spacing a satisfying a<O(1).

DOI
10.1103/physreva.38.3175

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The eigenvalue moment method for bosonic systems, recently developed by C. R. Handy and D. Bessis [Phys. Rev. Lett 55, 931 (1985)], is extended to discrete quantum mechanics. The relevant formalism is described and applied in the context of the discretized harmonic and sextic anharmonic potentials. Rapidly converging bounds to the associated ground-state energies are obtained for fixed lattice spacing a satisfying a<O(1).
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System-specific coherent states are constructed based on the formulation of supersymmetric quantum mechanics for arbitrary quantum systems. By regarding the superpotential as a generalized displacement variable, we identity the ground state of a quantum system as the minimizer of the supersymmetric Heisenberg uncertainty product. A special case is the ground state of the standard harmonic oscillator. One constructs standard coherent states by applying a shift operator to a 'fiducial function', taken as the ground state Gaussian. By analogy, we use the ground state for any other system as a new fiducial function, generating from its shifts new dynamically-adapted, overcomplete coherent states. The discretized system-specific coherent states can serve as a dynamically-adapted basis for bound state calculations. Accurate computational results for the Morse potential, the double well potential and the two-dimensional anharmonic oscillator systems demonstrate that the system-specific coherent states can provide rapidly-converging approximations for excited state energies and wave functions.
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There has been renewed interest in the exploitation of Barta's configuration space theorem (BCST) (Barta 1937 C. R. Acad. Sci. Paris 204 472) which bounds the ground-state energy, , by using any Φ lying within the space of positive, bounded, and sufficiently smooth functions, . Mouchet's (Mouchet 2005 J. Phys. A: Math. Gen. 38 1039 ) BCST analysis is based on gradient optimization (GO). However, it overlooks significant difficulties: (i) appearance of multi-extrema; (ii) inefficiency of GO for stiff (singular perturbation/strong coupling) problems; (iii) the nonexistence of a systematic procedure for arbitrarily improving the bounds within . These deficiencies can be corrected by transforming BCST into a moments' representation equivalent, and exploiting a generalization of the eigenvalue moment method (EMM), within the context of the well-known generalized eigenvalue problem (GEP), as developed here. EMM is an alternative eigenenergy bounding, variational procedure, overlooked by Mouchet, which also exploits the positivity of the desired physical solution. Furthermore, it is applicable to Hermitian and non-Hermitian systems with complex-number quantization parameters (Handy and Bessis 1985 Phys. Rev. Lett. 55 931, Handy et al 1988 Phys. Rev. Lett. 60 253, Handy 2001 J. Phys. A: Math. Gen. 34 5065 , Handy et al 2002 J. Phys. A: Math. Gen. 35 6359 ). Our analysis exploits various quasi-convexity/concavity theorems common to the GEP representation. We outline the general theory, and present some illustrative examples.