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Equidistant spectra of anharmonic oscillators
S. Yu. Dubov1, V. M. Eleonskii, N. E. Kulagin
1Research Institute of Physical Problems, Zelenograd, Moscow 103460, Russia.
Chaos (Woodbury, N.Y.)
|March 1, 1994
Summary
Researchers constructed anharmonic oscillator potentials and spectral-shift operators, generalizing Fok operators. This leads to equidistant excited state spectra and reveals connections to nonclassical orthogonal polynomials.
Area of Science:
- Quantum mechanics
- Mathematical physics
Background:
- Anharmonic oscillators are crucial in quantum mechanics for modeling various physical systems.
- Spectral-shift operators offer a powerful tool for analyzing quantum systems.
Purpose of the Study:
- To construct anharmonic oscillator potentials and corresponding spectral-shift operators.
- To analyze the properties of dynamic systems generated by third-degree spectral-shift operators.
- To establish the relationship between quantum system eigenvectors and nonclassical orthogonal polynomials.
Main Methods:
- Construction of representative anharmonic oscillator potentials.
- Development of spectral-shift operators as a generalization of Fok operators.
- Analysis of the Schrödinger problem for these potentials.
Main Results:
- An equidistant energy spectrum for excited states, separated by an energy gap from the ground state.
- Characterization of dynamic systems generated by third-degree spectral-shift operators.
- Establishment of a connection between Schrödinger problem eigenvectors and nonclassical orthogonal polynomials.
Conclusions:
- The constructed spectral-shift operators provide a novel approach to studying anharmonic oscillators.
- The findings reveal a deep connection between quantum mechanics and orthogonal polynomial theory.