Abstract
We investigated quantum states with continuous spectrum for a general time-dependent oscillator using invariant operator and unitary transformation methods together. The form of the transformed invariant operator by a unitary operator is the same as the Hamiltonian of the simple harmonic oscillator:I’ = p2/2 +ω 2 q 2/2. The fact thatω 2 of the transformed invariant operator is constant enabled us to investigate the system separately for three cases, whereω 2 > 0,ω 2 < 0, andω 2 = 0. The eigenstates of the system are discrete forω 2 > 0. On the other hand, forω 2 <− 0, the eigenstates are continuous. The time-dependent oscillators whose spectra of the wave function are continuous are not oscillatory. The wave function forω 2 < 0 is expressed in terms of the parabolic cylinder function. We applied our theory to the driven harmonic oscillator with strongly pulsating mass.
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