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A general dynamical invariant operator for three coupled time-dependentoscillators is derived. Although the obtained invariant operator satisfies theLiouville-von Neumann equation, its mathematical formula is somewhatcomplicated due to arbitrariness of time variations of parameters. Theparametric conditions required for formulating this invariant are definitelyspecified. By using the unitary transformation method, the invariant operatoris transformed to the one that corresponds to three independent simple harmonicoscillators. Inverse transformation of the well-known quantum solutionsassociated with such a simplified invariant enables us to identify quantumsolutions of the coupled original systems. These solutions are exact since wedo not use approximations not only in formulating the invariant operator but inthe unitary transformation as well. The invariant operator and itseigenfunctions provided here can be used to characterize quantum properties ofthe systems with various choices of the types of time-dependent parameters.

Formulation of general dynamical invariants and their unitary relations for time-dependent three coupled quantum oscillators