Eigenvalues of the two-dimensional Schr�dinger equation with nonseparable potentials

Taşeli, Hasan; Eid, R.

doi:10.1002/(sici)1097-461x(1996)59:3<183::aid-qua2>3.0.co;2-u

Cited by 17 publications

(22 citation statements)

References 14 publications

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“…Our results, see Tables I -IV, are in complete agreement with [6], [10] for D = 1 and superior considerably of those obtained for D > 1 and different g 2 , see e.g. [11], [12] and [13].…”

Section: The Approximant and Variational Calculationssupporting

confidence: 90%

“…In general, our results reproduce all known numerical ones that can be found for D = 1, 2, 3, see e.g. [12], [11] [18] and [19].…”

Section: The Approximant and Variational Calculationssupporting

confidence: 90%

“…For both equations (7) and (11) the Perturbation Theory (PT) in powers of the effective coupling constant λ can be developed; it generates the weak coupling expansion for ε ε(λ) = ∞ n=0 ε n λ n (13) and for the functions Y(v) and Z(u),…”

Section: Equationmentioning

confidence: 99%

See 2 more Smart Citations

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Valle

Turbiner

2020

Int. J. Mod. Phys. A

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In our previous paper I (del Valle-Turbiner, 2019) it was developed the formalism to study the general D-dimensional radial anharmonic oscillator with potential V (r) = 1 g 2V (gr). It was based on the Perturbation Theory (PT) in powers of g (weak coupling regime) and in inverse, fractional powers of g (strong coupling regime) in both r-space and in (gr)-space, respectively. As the result it was introduced -the Approximant -a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials V (r) = r 2 + g 2(m−1) r 2m , m = 2, 3, respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8-12 figures for any D = 1, 2, 3 . . . and g ≥ 0, while the relative deviation of the Approximant from the exact eigenfunction is less than 10 −6 for any r ≥ 0.

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“…Our results, see Tables I -IV, are in complete agreement with [6], [10] for D = 1 and superior considerably of those obtained for D > 1 and different g 2 , see e.g. [11], [12] and [13].…”

Section: The Approximant and Variational Calculationssupporting

confidence: 90%

“…In general, our results reproduce all known numerical ones that can be found for D = 1, 2, 3, see e.g. [12], [11] [18] and [19].…”

Section: The Approximant and Variational Calculationssupporting

confidence: 90%

See 1 more Smart Citation

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Valle

Turbiner

2020

Int. J. Mod. Phys. A

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“…In a recent article [13] by the same authors, it was shown numerically that the Dirichlet boundary value problem,…”

Section: Introductionmentioning

confidence: 99%

Converging bounds for the eigenvalues of multiminima potentials in two-dimensional space

Taşeli¹,

Eid²

1996

J. Phys. A: Math. Gen.

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The eigenvalue problem in L 2 (R 2) of Schrödinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls. It is shown that the Dirichlet and the von Neumann problems in L 2 () generate upper and lower bounds, respectively, to the eigenvalues of the unbounded system. To illustrate the method, rapidly converging two-sided bounds for the energy levels of two-and three-well oscillators are presented, using simple trigonometric basis functions.

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“…Thus, we expand the wave functions in terms of the trigonometric basis functions obeying Dirichlet boundary condition. Then we use the Rayleigh-Ritz variational scheme to optimize the domain of the basis functions which leads to highly accurate solutions [6][7][8][9][10]. …”

Section: Introductionmentioning

confidence: 99%

Accurate Energy Spectrum for the Quantum Yang-Mills Mechanics with Nonlinear Color Oscillations

Pedram

2014

Int J Theor Phys

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Yang-Mills theory as the foundation for quantum chromodynamics is a non-Abelian gauge theory with self-interactions between vector particles. Here, we study the Yang-Mills Hamiltonian with nonlinear color oscillations in the absence of external sources corresponding to the group SU (2). In the quantum domain, we diagonalize the Hamiltonian using the optimized trigonometric basis expansion method and find accurate energy eigenvalues and eigenfunctions for one and two degrees of freedom. We also compare our results with the semiclassical solutions.

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Eigenvalues of the two-dimensional Schr�dinger equation with nonseparable potentials

Cited by 17 publications

References 14 publications

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Converging bounds for the eigenvalues of multiminima potentials in two-dimensional space

Accurate Energy Spectrum for the Quantum Yang-Mills Mechanics with Nonlinear Color Oscillations

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