We study Schrödinger operators on R³ with finitely many concentric spherical –shell interactions. The operators are defined via quadratic forms and characterized by continuity across each shell together with the standard jump condition for the normal derivative. A particular feature of the present work is that the resolvent representation is obtained in a fully three–dimensional boundary integral formulation that allows for finitely many shells with surface strengths ⱼ that may vary along each shell and are assumed to be bounded real–valued functions, rather than relying on a reduction to radial one–dimensional problems. Using a boundary integral approach based on the free Green kernel and single–layer potentials, we derive an explicit resolvent formula for an arbitrary number of shells with bounded coupling strengths. This yields a concrete Kreĭn–type representation and a boundary operator whose noninvertibility characterizes the discrete spectrum, and it is compatible with partial–wave reduction under rotational symmetry. We then specialize to the two–shell case with constant couplings and obtain a detailed description of the negative spectrum. In particular, we prove that the ground state, when it exists, lies in the s–wave sector and derive an explicit secular equation for bound states. For large shell separation, each bound level approaches the corresponding single–shell level with exponentially small corrections, while a genuine tunneling splitting appears when the single–shell levels are tuned to coincide. As a simple calibration, we relate the two–shell parameters to representative core–shell quantum dot scales and identify the qualitative distinction between Type I and Type II configurations.
Masahiro Kaminaga (Mon,) studied this question.