Determinant Formulas for Scattering Matrices of Schrödinger Operators with Finitely Many Concentric $\delta$-Shells

We study stationary scattering for Schrödinger operators in ${\mathbb{R}}^3$ with finitely many concentric $\delta$–shell interactions of constant real strengths. Starting from the self–adjoint realization and the boundary resolvent formula for this model, we show that, after partial–wave reduction, the same finite-dimensional boundary matrices that arise in the resolvent formula also determine the channel scattering coefficients. More precisely, for each angular momentum $\mathrm{\ell }$, the channel coefficient $S_{\mathrm{\ell }}(k)$ satisfies $S_{\mathrm{\ell }}(k)=\det K_{\mathrm{\ell }}(k^2-i0)\mathrm{/}\det K_{\mathrm{\ell }}(k^2+i0)$ for almost every $k>0$, where $K_{\mathrm{\ell }}(z)=I_N+m_{\mathrm{\ell }}(z)\mathrm{\Theta }$ is the $\mathrm{\ell }$–th reduced boundary matrix. Thus, in each channel, the positive–energy scattering problem is reduced to a finite-dimensional matrix problem, and the scattering phase is recovered from $\det K_{\mathrm{\ell }}(k^2+i0)$. We then study the first nontrivial case of two concentric shells in the $s$–wave channel, where the interaction between the shells produces nontrivial threshold effects. We derive an explicit formula for $S_0(k)$ and analyze its behavior as $k\downarrow 0$. In the regular threshold regime, we obtain an explicit scattering length. We further identify a threshold–critical configuration characterized by the existence of a nontrivial zero–energy radial solution, regular at the origin, whose exterior constant term vanishes. In the corresponding nondegenerate exceptional case, the usual finite scattering length breaks down, and instead $S_0(k)\to -1$ as $k\downarrow 0$.