Paths to optimization in the multistate Rayleigh-Ritz variational method: Applications to the double-well quantum anharmonic oscillator

Bishop, R. F.; Flynn, Mary F.; Boscá, M. C.; Guardiola, R.

doi:10.1103/physreva.40.6154

Cited by 22 publications

(25 citation statements)

References 59 publications

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“…I. By the variational principle, our computed eigenenergies are upper bounds to the true eigenenergies [29,30,40]. Therefore, lower eigenenergy values always imply higher accuracy.…”

Section: Tight Binding Applied To the Current-mirror Circuitmentioning

confidence: 93%

“…(10). Note that an alternative route to this equation is application of the variational principle to ψ ng |H|ψ ng = E ψ ng |ψ ng [29]; the benefit of this viewpoint is that the eigenenergies thus obtained represent upper bounds to the true eigenenergies of the system [29,30]. Our analysis thus far has assumed a purely periodic potential, allowing for a direct analogy with the theory of tight binding as applied to solids.…”

Section: B Bloch Summation and The Generalized Eigenvalue Problemmentioning

confidence: 99%

“…Alternatively, one could optimize the harmonic lengths of a higher-lying basis state[29], which is a possible avenue for future research.…”

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confidence: 99%

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Variational tight-binding method for simulating large superconducting circuits

Weiss,

DeGottardi,

Koch

et al. 2021

Preprint

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We generalize solid-state tight-binding techniques for the spectral analysis of large superconducting circuits. We find that tight-binding states can be better suited for approximating the low-energy excitations than chargebasis states, as illustrated for the interesting example of the current-mirror circuit. The use of tight binding can dramatically lower the Hilbert space dimension required for convergence to the true spectrum, and allows for the accurate simulation of larger circuits that are out of reach of charge basis diagonalization.

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“…I. By the variational principle, our computed eigenenergies are upper bounds to the true eigenenergies [29,30,40]. Therefore, lower eigenenergy values always imply higher accuracy.…”

Section: Tight Binding Applied To the Current-mirror Circuitmentioning

confidence: 93%

Section: B Bloch Summation and The Generalized Eigenvalue Problemmentioning

confidence: 99%

See 1 more Smart Citation

Variational tight-binding method for simulating large superconducting circuits

Weiss,

DeGottardi,

Koch

et al. 2021

Preprint

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“…in which x ∈ (−∞, +∞) and g ∈ [0, ∞). The problem of finding the eigenvalues of Hamiltonian (81) is a challenge for any analytical method, although there are several numerical techniques calculating the eigenvalues with reasonable accuracy [38][39][40][41][42][43]. It is especially difficult to calculate the lowest energy levels.…”

Section: B One-dimensional Oscillatormentioning

confidence: 99%

Self-similar interpolation in quantum mechanics

1998

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An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive interpolation formulae valid in the whole range of parameters of considered physical quantities, the self-similar renormalization procedure is complimented here by boundary conditions which define control functions guaranteeing correct asymptotic behaviour in the vicinity of boundary points. To emphasize the generality of the approach, it is illustrated by different problems that are typical for quantum mechanics, such as anharmonic oscillators, doublewell potentials, and quasiresonance models with quasistationary states. In addition, the nonlinear Schrödinger equation is considered, for which both eigenvalues and wave functions are constructed.02.30.Lt, 02.30.Mv, 03.65 Ge

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“…The method constructs an approximation using the eigenfunctions associated with the smaller eigenvalues of an eigenvalue problem under a variational framework where the eigenfunctions serve as the reduced‐order bases. The Rayleigh–Ritz method has been applied to structural dynamics , compressible flow , and quantum chemistry , among others. In the past decades, the projection‐based MOR techniques have been widely applied to numerous areas, such as structural dynamics , turbulence flows , fluid dynamics , aeroelastic‐coupled fluid‐structure system and structural subsystem , electro‐thermal microelectromechanical system , and biological image analysis , to name just a few.…”

Section: Introductionmentioning

confidence: 99%

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

Lee

Chen

2013

Numerical Meth Engineering

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SUMMARYModel order reduction for molecular dynamics (MD) systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi-atomic interactions in high dimensions. In the present work, we introduce a proper orthogonal decomposition-based method in conjunction with the radial basis function (RBF) approximation of the nonlinear and nonlocal potential energies and inter-atomic forces for MD systems. This approach avoids coordinate transformation between the physical and reduced-order coordinates, and allows the potentials and inter-atomic forces to be calculated directly in the reduced-order space. The RBFapproximated potential energies and inter-atomic forces in the reduced-order space are discretized on the basis of the Smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method. The good approximation properties of RBFs in interpolating scattered data make them ideal candidates for the reduced-order approximation of MD inter-atomic force calculations. The proposed approach is validated by performing the reduced-order simulations of DNA molecules under various external loadings. Copyright

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Paths to optimization in the multistate Rayleigh-Ritz variational method: Applications to the double-well quantum anharmonic oscillator

Abstract: Both single-well and multiwell one-dimensional anharmonic oscillators have provided an extremely fruitful testing ground for various modern techniques of microscopic quantum many-body theory and quantum field theory. They have served both as (0+1)-dimensional field-theory models

Cited by 22 publications

References 59 publications

Variational tight-binding method for simulating large superconducting circuits

Variational tight-binding method for simulating large superconducting circuits

Self-similar interpolation in quantum mechanics

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

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