  We propose a reconfigurable intelligent surface (RIS)-assisted underlay
spectrum sharing system, in which a RIS-assisted secondary network shares the
spectrum licensed for a primary network. The secondary network consists of a
secondary source (SS), an RIS, and a secondary destination (SD), operating in a
Rician fading environment. We study the performance of the secondary network
while considering a peak power constraint at the SS and an interference power
constraint at the primary receiver (PR). Initially, we characterize the
statistics of the signal-to-noise ratio (SNR) of the RIS-assisted secondary
network by deriving novel analytical expressions for the cumulative
distribution function (CDF) and probability density function (PDF) in terms of
the incomplete H-function. Building upon the SNR statistics, we analyze the
outage probability, ergodic capacity, and average bit error rate, subsequently
deriving novel exact expressions for these performance measures. Furthermore,
we obtain novel asymptotic expressions for the performance measures of interest
when the peak power of the SS is high. Finally, we conduct exhaustive
Monte-Carlo simulations to confirm the correctness of our theoretical analysis.
