Dynamic polarizability of low-dimensional excitons

Excitons in low-dimensional materials behave mathematically as confined hydrogen atoms. An appealing unified description of confinement in quantum wells or wires, etc., is found by restricting space to a fractional dimension $1lD\ensuremath{\le}3$ serving as an adjustable parameter. We compute the dynamic polarizability of D-dimensional excitons in terms of discrete and continuum oscillator strengths. Analyzing exact sum rules, we show that continuum contributions are increasingly important in low dimensions. The dynamical responses of excitons in various dimensions are compared. Finally, an exact and compact closed-form expression for the dynamic polarizability is found. This completely general formula takes D as input and provides exact results for arbitrary frequency.