Perelman's $\lambda$-functional on manifolds with conical singularities

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.