Vector analysis package

<<VectorAnalysis`

Laplacian[f[Xx,Yy,Zz]]

Laplacian[f[Rr, Ttheta, PPhi], Spherical]

Laplacian[f[Rr, Ttheta, PPhi], Spherical]//FullSimplify

On the unit sphere, the spherical harmonics are eigenfunctions of the Laplacian

SphericalHarmonicY[1,0,Ttheta,Pphi]

Laplacian[SphericalHarmonicY[1,0,Ttheta,Pphi],Spherical]

I.e., this is - 1/2 times Y[1, 0] with Rr = 1.

SphericalHarmonicY[2,1,Ttheta,Pphi]

Laplacian[SphericalHarmonicY[2,1,Ttheta,Pphi],Spherical]//FullSimplify

See?  Now try to figure out the general formula for the eigenvalues of the Ylm.