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

Abstract

We study stationary scattering for Schr\"odinger 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 $\ell$, the channel coefficient $S_\ell(k)$ satisfies $S_\ell(k)=\det K_\ell(k^2-i0)/\det K_\ell(k^2+i0)$ for almost every $k>0$, where $K_\ell(z)=I_N+m_\ell(z)\Theta$ is the $\ell$--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 $\det K_\ell(k^2+i0)$. 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 $k\downarrow0$. 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 $k\downarrow0$.