Dynamic polarizability of low-dimensional excitons

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