  In this paper, it is shown that some new phenomenon related to the spectra of
Toeplitz operators with bounded harmonic symbols on the Bergman space. On the
one hand, we prove that the spectrum of the Toeplitz operator with symbol
${\bar{z}+p}$ is always connected for every polynomial $p$ with degree less
than $3$. On the other hand, we show that for each integer $k$ greater than
$2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the
Toeplitz operator with symbol ${\bar{z}+p}$ has at least one isolated point but
has at most finitely many isolated points. Then these results are applied to
obtain a new class of non-hyponormal Toeplitz operators with bounded harmonic
symbols on the Bergman space for which Weyl's theorem holds.
