Classical Limit on Quantum Mechanics for Unbounded Observables

This dissertation is divided into two parts. In Part I of this dissertation--- On the Classical Limit of Quantum Mechanics, we extend a method introduced by Hepp in 1974 for studying the asymptotic behavior of quantum expectations in the limit as Plank's constant (h) tends to zero. The goal is to allow for unbounded observables which are (non-commutative) polynomial functions of the position and momentum operators. [This is in contrast to Hepp's original paper where the "observables" were, roughly speaking, required to be bounded functions of the position and momentum operators.] As expected the leading order contributions of the quantum expectations come from evaluating the "symbols" of the observables along the classical trajectories while the next order contributions (quantum corrections) are computed by evolving the h=1 observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.;Part II of the dissertation --- Powers of Symmetric Differential Operators is devoted to operator theoretic properties of a class of linear symmetric differential operators on the real line. In more detail, let L and L˜ be a linear symmetric differential operator with polynomial coefficients on L2 (m) whose domain is the Schwartz test function space, S. We study conditions on the polynomial coefficients of L and L˜ which implies operator comparison inequalities of the form (L˜+C˜r ≤ Cr (L¯+C) r for all 0 ≤ r < infinity. These comparison inequalities (along with their generalizations allowing for the parameter h>0 in the coefficients) are used to supply a large class of Hamiltonian operators which verify the assumptions needed for the results in Part I of this dissertation.