The Schrödinger equation for bound state systems in the oscillator representation

Dineykhan, M.; Efimov, G. V.

doi:10.1016/0034-4877(96)83626-3

Cited by 28 publications

(54 citation statements)

References 17 publications

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“…The scaling self-similarity is what one usually keeps in mind when mentioning this property. However, this type of self-similarity is just a particular trivial case of the relation (33). The latter for the endomorphism y k , together with the unitary element (25), defines the semigroup properties…”

Section: Self-similar Approximation Theorymentioning

confidence: 99%

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Self-similar structures and fractal transforms in approximation theory

Yukalov

Yukalova

2002

Chaos, Solitons & Fractals

112

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An overview is given of the methods for treating complicated problems without small parameters, when the standard perturbation theory based on the existence of small parameters becomes useless. Such complicated problems are typical of quantum physics, many-body physics, physics of complex systems, and various aspects of applied physics and applied mathematics. A general approach for dealing with such problems has been developed, called Self-Similar Approximation Theory. A concise survey of the main ideas of this approach is presented, with the emphasis on the basic notion of group self-similarity. The techniques are illustrated by examples from quantum field theory.

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Section: Self-similar Approximation Theorymentioning

confidence: 99%

“…Many problems of quantum mechanics, statistical mechanics, condensed matter physics, and quantum field theory are successively treated by this optimized approach. Not to list numerous applications, let us cite a couple of recent reviews [33,34].…”

Section: Optimized Perturbation Theorymentioning

confidence: 99%

Self-similar structures and fractal transforms in approximation theory

Yukalov

Yukalova

2002

Chaos, Solitons & Fractals

112

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“…As is known, the more used methods are diagonalization [2,3], quasiclassical approach [4], continuous analogue of Newton's method [5], different versions of perturbation theory [6], normal form method [7,8,9,10,11], finite-element method [12], 1/N expansion [13], oscillator representation [14], variational and operational methods [15,16,17], simplectic method [18] and etc.…”

Section: Introductionmentioning

confidence: 99%

A Symbolic-Numeric Approach for Solving the Eigenvalue Problem for the One-Dimensional Schrödinger Equation

Belyaeva

Chekanov

Гусев

et al. 2006

Computer Algebra in Scientific Computing

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Abstract.A general scheme of a symbolic-numeric approach for solving the eigenvalue problem for the one-dimensional Shrödinger equation is presented. The corresponding algorithm of the developed program EWA using a conventional pseudocode is described too. With the help of this program the energy spectra and the wave functions for some Schrödinger operators such as quartic, sextic, octic anharmonic oscillators including the quartic oscillator with double well are calculated.

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“…Our analysis is based on the method of oscillator representation [11] that has already been used advantageously in calculations of the energy spectra of various systems. In section 1, the interaction Hamiltonian is determined.…”

Section: Introductionmentioning

confidence: 99%

Stark Splitting of the Energy Spectrum in a Two-Electron Quantum Dot

2005

Self Cite

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A two-electron quantum dot model assuming the existence of the image charge is formulated in the present paper. Based on this model, the effective electron confinement potential is calculated by the self-consistent method. It is established that the electron confinement potential in the quantum dot has both symmetric and antisymmetric parts. An analysis of this potential demonstrates that the tunnel electron transition in the quantum dot can occur only in the direction perpendicular to the film plane.

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The Schrödinger equation for bound state systems in the oscillator representation

Cited by 28 publications

References 17 publications

Self-similar structures and fractal transforms in approximation theory

Self-similar structures and fractal transforms in approximation theory

A Symbolic-Numeric Approach for Solving the Eigenvalue Problem for the One-Dimensional Schrödinger Equation

Stark Splitting of the Energy Spectrum in a Two-Electron Quantum Dot

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