scholarly journals Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures

2021 ◽  
Vol 3 (3) ◽  
pp. 458-472
Author(s):  
Kevin Zelaya ◽  
Oscar Rosas-Ortiz
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Exactly Solvable ◽  
Point Transformations

We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectra, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola–Kanai model as a particular case. Explicit solutions are given in both the classical and quantum pictures.

COMPLETE EXACT QUANTUM STATES OF THE GENERALIZED TIME-DEPENDENT HARMONIC OSCILLATOR

2004 ◽  
Vol 18 (24) ◽  
pp. 1267-1274 ◽  
Author(s):  
I. A. PEDROSA
Keyword(s):  
General Solution ◽  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Continuous Spectrum ◽  
Invariant Operator ◽  
Quantum States ◽  
Time Dependent ◽  
Quadratic Invariants ◽  
Generalized Time ◽  
Discrete And Continuous Spectrum

By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.

scholarly journals The complete symmetry group of a forced harmonic oscillator

1980 ◽  
Vol 22 (1) ◽  
pp. 12-21 ◽  
Author(s):  
P. G. L. Leach
Keyword(s):  
Harmonic Oscillator ◽  
Symmetry Group ◽  
Lie Groups ◽  
Time Dependent ◽  
One Dimensional ◽  
Complete Symmetry ◽  
Point Transformations ◽  
The One ◽  
Dependent Point ◽  
Complete Symmetry Group

AbstractThe complete symmetry group of a forced harmonic oscillator is shown to be Sl(3, R) in the one-dimensional case. Approaching the problem through the Hamiltonian invariants and the method of extended Lie groups, the method used is that of time-dependent point transformations. The result applies equally well to the forced repulsive oscillator and a particle moving under the influence of a coordinate-free force. The generalization to na-dimensional systems is discussed.

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