  For a certain domain $\Omega$ in the Sierpinski gasket $\mathcal{SG}$ whose
boundary is a line segment, a complete description of the eigenvalues of the
Laplacian, with an exact count of dimensions of eigenspaces, under the
Dirichlet and Neumann boundary conditions is presented. The method developed in
this paper is a weak version of the spectral decimation method due to Fukushima
and Shima, since for a lot of "bad" eigenvalues the spectral decimation method
can not be used directly. Let $\rho^0(x)$, $\rho^\Omega(x)$ be the eigenvalue
counting functions of the Laplacian associated to $\mathcal{SG}$ and $\Omega$
respectively. We prove a comparison between $\rho^0(x)$ and $\rho^\Omega(x)$
says that $ 0\leq \rho^{0}(x)-\rho^\Omega(x)\leq C x^{\log2/\log 5}\log x$ for
sufficiently large $x$ for some positive constant $C$. As a consequence,
$\rho^\Omega(x)=g(\log x)x^{\log 3/\log 5}+O(x^{\log2/\log5}\log x)$ as
$x\rightarrow\infty$, for some (right-continuous discontinuous) $\log
5$-periodic function $g:\mathbb{R}\rightarrow \mathbb{R}$ with
$0<\inf_{\mathbb{R}}g<\sup_\mathbb{R}g<\infty$. Moreover, we explain that the
asymptotic expansion of $\rho^\Omega(x)$ should admit a second term of the
order $\log2/\log 5$, that becomes apparent from the experimental data. This is
very analogous to the conjectures of Weyl and Berry.
