An asymptotic expansion for energy eigenvalues of anharmonic oscillators

Gaudreau, Philippe; Slevinsky, Richard M.; Safouhi, Hassan

doi:10.1016/j.aop.2013.07.001

Cited by 21 publications

(12 citation statements)

References 21 publications

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“…In [39], Gaudreau et al derive a reversed WKB series for the eigenvalues of the Schrödinger equation on the real line with the quartic anharmonic potential V (x) = ωx 2 + x 4 of the form:…”

Section: Asymptotics Of the Spectra Of Confined Quantum Anharmonic Osmentioning

confidence: 99%

On symmetrizing the ultraspherical spectral method for self-adjoint problems

Aurentz

Slevinsky

2020

Journal of Computational Physics

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A mechanism is described to symmetrize the ultraspherical spectral method for self-adjoint problems. The resulting discretizations are symmetric and banded. An algorithm is presented for an adaptive spectral decomposition of self-adjoint operators. Several applications are explored to demonstrate the properties of the symmetrizer and the adaptive spectral decomposition.Polynomial approximation theory suggests that we should seek to numerically represent the variable coefficients in Eq. (1) as Chebyshev polynomial expansions [18]. An adaptation of the fast multipole method (FMM) [19] accelerates the Chebyshev-Legendre [20], ultraspherical-ultraspherical [21], and Jacobi-Jacobi [22, §3.3] connection problems to linear complexity in the degree. Provided the Jacobi parameters are not too large, the acceleration of the connection problem enables Jacobi polynomial expansions with nearly the same rapidity as Chebyshev polynomial expansions. Thus in the problems in this paper, the variable coefficients in Eq. (1) are in fact polynomials, but in practice they could very well be good numerical approximations to functions in a particular space. How such approximations, in particular of the leading coefficient p N , affect the spectrum is beyond the scope of this report. A well-known model problemIt is instructive to begin with a well-known model problem for illustrative purposes. 1 In exceptional circumstances, such as when B is zero apart from the first few columns and the nonzero part prepended to L renders the block operator discretization self-adjoint, symmetry may still be within reach. Many common boundary conditions, however, are genuinely infinite-dimensional. This destruction of symmetry has real consequences for the spectra of finite truncations. Whereas Chebyshev and Legendre second-order differentiation matrices by collocation have real spectra [9], it is easy to show by counter-example that finite truncations of ultraspherical discretizations of self-adjoint operators may produce complex spectra.

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Section: Asymptotics Of the Spectra Of Confined Quantum Anharmonic Osmentioning

confidence: 99%

On symmetrizing the ultraspherical spectral method for self-adjoint problems

Aurentz

Slevinsky

2020

Journal of Computational Physics

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“…The analytic structure of the renormalized strong coupling expansion for anharmonic oscillators has been thoroughly investigated . Moreover, an asymptotic expansion for the energy eigenvalues of anharmonic oscillators has been obtained using the Wentzel–Kramers–Brillouin (WKB) theory and series reversion . Recently, highly accurate energy eigenvalues for anharmonic oscillators have computed using the Sinc collocation method combined with the double exponential transformation …”

Section: Introductionmentioning

confidence: 99%

Asymptotic Functional Form Preservation Method for the Perturbation Theory: Application to Anharmonic Oscillators

Chou

2016

J Chinese Chemical Soc

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The asymptotic functional form preservation method is developed for the perturbation theory to obtain the energy eigenvalues of anharmonic oscillators. The conventional energy perturbative series expansion for the anharmonic oscillator is strongly divergent even if the anharmonicity is small. Employing a transformation containing an unphysical parameter, we analytically continue this series expansion into a new series expansion applicable to all the range of the perturbation parameter. The unphysical parameter is determined by the principle of minimal sensitivity. This new series expansion is reduced to the conventional energy perturbative series expansion for small anharmonicity, and it preserves the correct asymptotic functional form when the perturbation parameter tends to infinity. Then, we use the full‐range energy series expansion to calculate the energy eigenvalues of the anharmonic oscillator. In addition to excellent energy eigenvalues obtained for the oscillator with small and strong anharmonicity, accurate energy eigenvalues can be obtained using the full‐range energy series expansion when the perturbation parameter tends to infinity.

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“…For more details on the application of the SCM, we refer the readers to [9]. As in [39], we let δ (l) j,k denote the l th Sinc differentiation matrix with unit mesh size:…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

“…In [9], we applied the DESCM to obtain an approximation to the eigenvalues λ of equation (5). We initially applied Eggert et al's transformation to equation (5) since it was shown that the proposed change of variable results in a symmetric discretized system when using the Sinc collocation method [37].…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

“…By evaluating the resulting approximation at the Sinc collocation points separated by a fixed mesh size h, one obtains a matrix eigenvalue problem or generalized matrix eigenvalue problem for which the eigenvalues are approximations to the eigenvalues of the SL operator. In [9], we used the double exponential Sinc collocation method (DESCM) to compute the eigenvalues of singular Sturm-Liouville boundary value problems. The DESCM leads to a generalized eigenvalue problem where the matrices are symmetric and positive-definite.…”

Section: Introductionmentioning

confidence: 99%

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Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Gaudreau

Safouhi

2016

EPJ Web of Conferences

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Abstract. Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1 N+1 of all components need to be computed and stored in order to obtain all eigenvalues, where 2N + 1 corresponds to the dimension of the eigensystem. We applied our result to the Schrödinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.

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An asymptotic expansion for energy eigenvalues of anharmonic oscillators

Cited by 21 publications

References 21 publications

On symmetrizing the ultraspherical spectral method for self-adjoint problems

On symmetrizing the ultraspherical spectral method for self-adjoint problems

Asymptotic Functional Form Preservation Method for the Perturbation Theory: Application to Anharmonic Oscillators

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

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