UC San Diego

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 ($\hbar$) 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 $\hbar=1$ 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 $\tilde{L}$ be a linear symmetric differential operator with polynomial coefficients on $L^{2}\left(m \right) $ whose domain is the Schwartz test function space, $\mathcal{S}.$ We study conditions on the polynomial coefficients of $L$ and $\tilde{L}$ which implies operator comparison inequalities of the form $\left( \overline{\tilde{L}}+\tilde{C}\right) ^{r}\leq C_{r}\left( \bar{L}+C\right) ^{r}$

for all $0\leq r<\infty.$ These comparison inequalities (along with their generalizations

allowing for the parameter $\hbar>0$ 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.