Exact solutions of the sextic oscillator from the bi-confluent Heun equation

Lévai, G.; Ishkhanyan, А. М.

doi:10.1142/s0217732319501347

Cited by 20 publications

(16 citation statements)

References 58 publications

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“…The quartic and sextic oscillators both belong to the class of potentials whose exact solutions are given by Heun's special function [37]. In our harmonic basis, the nonzero Hamiltonian matrix elements are

H_{n, n + 2 k} = \sqrt{n_{2 k}} h_{k}^{(6)} / 48 ω^{3}

where

h_{3}^{(6)} = 1

and

\begin{matrix} lefttrue \\ h_{0}^{((), 6)} = (2 n + 1) [10 n (n + 1) + 3 (4 ω^{4} + 5)], \\ h_{1}^{((), 6)} = 3 [5 n (n + 3) - 4 ω^{4} + 15], \\ h_{2}^{((), 6)} = 3 (2 n + 5), \end{matrix}

that is, they go one more step away from the diagonal.…”

Section: Resultsmentioning

confidence: 99%

Uncommonly accurate energies for the general quartic oscillator

Okun

Burke

2020

Int J of Quantum Chemistry

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Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10−34, but few non‐trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x2 and x4) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next two terms in the asymptotic expansion. We give analytic formulas for expansion in up to three even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond.

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H_{n, n + 2 k} = \sqrt{n_{2 k}} h_{k}^{(6)} / 48 ω^{3}

where

h_{3}^{(6)} = 1

and

\begin{matrix} lefttrue \\ h_{0}^{((), 6)} = (2 n + 1) [10 n (n + 1) + 3 (4 ω^{4} + 5)], \\ h_{1}^{((), 6)} = 3 [5 n (n + 3) - 4 ω^{4} + 15], \\ h_{2}^{((), 6)} = 3 (2 n + 5), \end{matrix}

that is, they go one more step away from the diagonal.…”

Section: Resultsmentioning

confidence: 99%

Uncommonly accurate energies for the general quartic oscillator

Okun

Burke

2020

Int J of Quantum Chemistry

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“…where n = 4, 6, we undertake to solve the corresponding equations ( 8) and (9). It is evident that we have the commutation relation…”

Section: General Features Of Canonical Hamiltonian Dynamicsmentioning

confidence: 99%

“…To the best of our knowledge, such computation has not been undertaken so far. Only the standard harmonic oscillator has been studied in details since the early days of the formulation of Moyal and Groenevold, thanks to the possibility to solve fully the two equations ( 8) and (9).…”

Section: General Features Of Canonical Hamiltonian Dynamicsmentioning

confidence: 99%

Moyal equation—Wigner distribution functions for anharmonic oscillators

Truong

2021

Journal of Mathematical Physics

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We investigate the structure of Wigner distribution functions of energy eigenstates of quartic and sextic anharmonic oscillators. The corresponding Moyal equations are shown to be solvable, revealing new properties of Schrödinger eigenfunctions of these oscillators. In particular, they provide alternative approaches to their determination and corresponding eigenenergies, away from the conventional differential equation method, which usually relies on the not so widely known bi-and tri-confluent Heun functions.

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“…The quartic and sextic oscillators both belong to the class of potentials whose exact solutions are given by Heun's special function [29]. In our harmonic basis, the nonzero Hamiltonian matrix elements are…”

Section: Sextic Oscillatormentioning

confidence: 99%

Uncommonly accurate energies for the general quartic oscillator

Okun¹,

Burke²

2020

Preprint

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Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10 −34 , but few non-trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x 2 and x 4 ) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next 2 terms in the asymptotic expansion. We give analytic formulas for expansion in up to 3 even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond.

show abstract

Exact solutions of the sextic oscillator from the bi-confluent Heun equation

Cited by 20 publications

References 58 publications

Uncommonly accurate energies for the general quartic oscillator

Uncommonly accurate energies for the general quartic oscillator

Moyal equation—Wigner distribution functions for anharmonic oscillators

Uncommonly accurate energies for the general quartic oscillator

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