  In this paper, we prove that on a compact manifold with isolated conical
singularity the spectrum of the Schr\"odinger operator $-4\Delta+R$ consists of
discrete eigenvalues with finite multiplicities, if the scalar curvature $R$
satisfies a certain condition near the singularity. Moreover, we obtain an
asymptotic behavior for eigenfunctions near the singularity. As a consequence
of these spectral properties, we extend the theory of the Perelman's
$\lambda$-functional on smooth compact manifolds to compact manifolds with
isolated conical singularities.
