  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 1
< D <= 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.
