Parametric Resonance in a Time-Dependent Harmonic Oscillator
<p>In this paper, we study the phenomenon of appearance of new resonances in a timedependent harmonic oscillator under an oscillatory decreasing force. The studied equation belongs to the class of adiabatic oscillators and arises in connection with the spectral problem for the one-dimensional...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2013-01-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | http://mais-journal.ru/jour/article/view/192 |
| _version_ | 1797967217077256192 |
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| author | P. N. Nesterov |
| author_facet | P. N. Nesterov |
| author_sort | P. N. Nesterov |
| collection | DOAJ |
| description | <p>In this paper, we study the phenomenon of appearance of new resonances in a timedependent harmonic oscillator under an oscillatory decreasing force. The studied equation belongs to the class of adiabatic oscillators and arises in connection with the spectral problem for the one-dimensional Schr¨odinger equation with Wigner–von Neumann type potential. We use a specially developed method for asymptotic integration of linear systems of differential equations with oscillatory decreasing coefficients. This method uses the ideas of the averaging method to simplify the initial system. Then we apply Levinson’s fundamental theorem to get the asymptotics for its solutions. Finally, we analyze the features of a parametric resonance phenomenon. The resonant frequencies of perturbation are found and the pointwise type of the parametric resonance phenomenon is established. In conclusion, we construct an example of a time-dependent harmonic oscillator (adiabatic oscillator) in which the parametric resonances, mentioned in the paper, may occur.</p> |
| first_indexed | 2024-04-11T02:26:37Z |
| format | Article |
| id | doaj.art-1eef635f17a04652840d4ea46a3e85a3 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-11T02:26:37Z |
| publishDate | 2013-01-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-1eef635f17a04652840d4ea46a3e85a32023-01-02T22:28:39ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172013-01-01203528186Parametric Resonance in a Time-Dependent Harmonic OscillatorP. N. Nesterov0Ярославский государственный университет им. П.Г. Демидова<p>In this paper, we study the phenomenon of appearance of new resonances in a timedependent harmonic oscillator under an oscillatory decreasing force. The studied equation belongs to the class of adiabatic oscillators and arises in connection with the spectral problem for the one-dimensional Schr¨odinger equation with Wigner–von Neumann type potential. We use a specially developed method for asymptotic integration of linear systems of differential equations with oscillatory decreasing coefficients. This method uses the ideas of the averaging method to simplify the initial system. Then we apply Levinson’s fundamental theorem to get the asymptotics for its solutions. Finally, we analyze the features of a parametric resonance phenomenon. The resonant frequencies of perturbation are found and the pointwise type of the parametric resonance phenomenon is established. In conclusion, we construct an example of a time-dependent harmonic oscillator (adiabatic oscillator) in which the parametric resonances, mentioned in the paper, may occur.</p>http://mais-journal.ru/jour/article/view/192гармонический осцилляторпеременная частотарезонансметод усредненияасимптотика |
| spellingShingle | P. N. Nesterov Parametric Resonance in a Time-Dependent Harmonic Oscillator Моделирование и анализ информационных систем гармонический осциллятор переменная частота резонанс метод усреднения асимптотика |
| title | Parametric Resonance in a Time-Dependent Harmonic Oscillator |
| title_full | Parametric Resonance in a Time-Dependent Harmonic Oscillator |
| title_fullStr | Parametric Resonance in a Time-Dependent Harmonic Oscillator |
| title_full_unstemmed | Parametric Resonance in a Time-Dependent Harmonic Oscillator |
| title_short | Parametric Resonance in a Time-Dependent Harmonic Oscillator |
| title_sort | parametric resonance in a time dependent harmonic oscillator |
| topic | гармонический осциллятор переменная частота резонанс метод усреднения асимптотика |
| url | http://mais-journal.ru/jour/article/view/192 |
| work_keys_str_mv | AT pnnesterov parametricresonanceinatimedependentharmonicoscillator |