Abstract
A framework for the momentum-resonance formulation of Lewis and Leach [Ann. Phys. (NY) 164, 47 (1985)] that casts new light into the nature of exact, explicitly time-dependent invariants for one-dimensional, time-dependent potentials and produces additional examples of such invariants is presented. The momentum-resonance formulation postulates that the invariant be a rational function of momentum with simple poles, which are called momentum resonances. It is shown that an invariant of resonance type can be written as a functional of the potential in terms of the solution of a system of linear algebraic equations; and a single necessary and sufficient condition for a potential to admit an invariant of resonance type is obtained. These results are obtained by reformulating the problem in terms of a set of discrete moments that satisfy two separate recursion formulas. Invariants for new time-dependent potentials can be obtained and previously known invariants are recovered.
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