UCLA

In this thesis, we examine aspects of non-self-adjoint (NSA) operators using the theory of microlocal analysis in exponentially weighted spaces of holomorphic functions on $\C^n$. We present four main results. The first theorem, which is the result of joint work with L. Coburn, M. Hitrik, and J. Sj\"{o}strand, establishes a boundedness criterion for a class of Toeplitz operators acting on Bargmann spaces with quadratic weights. The Toeplitz operators that this result applies to have symbols of the form $\exp{Q(z)}$, where $Q(z)$ is an inhomogenous quadratic polynomial on $\C^n$. The second and third results of this thesis establish properties of solutions of time-dependent Schr\"{o}dinger equations on $\R^n$ with NSA quadratic Hamiltonians. More specifically, these results pertain to solutions $u=u(t,x)$ of the initial value problem $(\p_t + q^w(x,D))u(t,x) = 0$, $x \in \R^n$, $t \ge 0$, on $\R^n$, where the initial data $u|_{t=0} = u_0$ is a tempered distribution on $\R^n$ and $q^w(x,D)$ is the Weyl quantization of a complex-valued quadratic form on the phase space $\R^{2n}$ with non-negative real part $\textrm{Re} \, q \ge 0$. Our second result characterizes the propagation in time of global analytic singularities of initial data $u_0$ by this evolution, and our third result establishes $L^p$-bounds for the evolution semigroup $e^{-tq^w(x,D)}$ in the short $0