This paper presents a comprehensive spectral analysis of the Dirichlet-to-Neumann operator associated with the Laplace equation in thin spherical shells in \(\mathbb {R}^3\) . Using spherical harmonic expansions and separation of variables, we derive an explicit spectral representation that diagonalizes the operator with respect to the spherical harmonic basis. We establish sharp continuity bounds in fractional Sobolev spaces on the sphere, proving explicit norm estimates in \(\mathcal {L}(\hbox {H}^{1/2}(\mathbb {S}^2)\) , \(\hbox {H}^{-1/2}(\mathbb {S}^2))\) . For the thin shell regime \(\delta \ll R\) , we develop rigorously justified first- and third-order asymptotic expansions with precise error estimates in operator norm after a frequency cut-off. The spectral properties of the operator are fully characterized, including self-adjointness, dissipativity, and high-frequency asymptotics. Our results provide both theoretical insights into boundary operators in singular geometries and practical approximation formulas for computational applications in domain decomposition methods and thin-layer modeling.
