| Summary: | Harmonic oscillators with multiple abrupt jumps in their frequencies have been investigated by several authors during the last decades. We investigate the dynamics of a quantum harmonic oscillator with initial frequency <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mn>0</mn></msub></semantics></math></inline-formula>, which undergoes a sudden jump to a frequency <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mn>1</mn></msub></semantics></math></inline-formula> and, after a certain time interval, suddenly returns to its initial frequency. Using the Lewis–Riesenfeld method of dynamical invariants, we present expressions for the mean energy value, the mean number of excitations, and the transition probabilities, considering the initial state different from the fundamental. We show that the mean energy of the oscillator, after the jumps, is equal or greater than the one before the jumps, even when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo><</mo><msub><mi>ω</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>. We also show that, for particular values of the time interval between the jumps, the oscillator returns to the same initial state.
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