Harmonic Analysis of the Semi-Classical Limit

Abstract

In the harmonic analysis of quantum-dynamical notions a method is presented that already in a simple perturbation-setting can successfully be used in order to obtain pointmechanical laws under the so-called classical nonrelativistic limit. In quantum mechanics a state sometimes is developed in terms of two-parameter families of unitary operators applied on the Hilbert space of absolutely square integrable functions in R3. Through a similar procedure one can introduce such two-parameter families of the unitary operators U(r,s) and Q(r,s) for all positive real values of r and s that U Q will represent a semigroup generated by a selfadjoint extension of the Laplace operator. In certain dynamical integrals one can moreover evaluate consistent limits when s →∞. By correspondence relations one may establish an isomorphy between the structure presented here and quantum dynamics. The existence of classical limits implies that s →∞. Reversely, s →∞ does not uniquely determine physical limits, and it is therefore necessary to specify the exact meanings of the classical and the semi-classical limits.