An explicit asymptotic formula is obtained for the norms of the spectral projections of the non-self-adjoint harmonic oscillator $H$. It is deduced that the spectral expansion of $\rme^{-Ht}$ is norm convergent if and only if $t$ is greater than a certain explicit positive constant.

We study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets, typical examples being the Meixner polynomials and the Charlier polynomials. Following ideas of Rakhmanov, Dragnev and Saff, weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials, which can be solved explicitly. We give formulas for the contracted zero distributions, the $n$th root asymptotics and the asymptotics of the largest zeros.

Using potential theoretic methods and exploiting the connection with eigenvalues of Toeplitz matrices, we investigate the limiting behaviour of zeros of Faber polynomials generated by a Laurent series. Our results build upon fundamental work of J. L. Ullman. For example, we show that if E is a compact set with simply connected complement and connected interior whose boundary is either (i) not a piecewise analytic curve or (ii) a piecewise analytic curve but with a singularity other than an outward cusp, then the equilibrium distribution for E is a limit measure of the sequence of normalized zero counting measures for the Faber polynomials associated with E.