Schr\"odinger operators with concentric $\delta$--shell interactions

TL;DR

This paper analyzes Schr"odinger operators with multiple concentric $\

Contribution

It provides an explicit resolvent formula and spectral analysis for Schr"odinger operators with multiple spherical $\

Findings

Explicit resolvent representation for multi-shell interactions.

Detailed description of the negative spectrum for two-shell systems.

Identification of tunneling effects and spectral behavior with shell separation.

Abstract

We study Schr\"odinger operators on $\mathbb R^3$ with finitely many concentric spherical $\delta$-shell interactions. The operators are defined by the quadratic form method and are described by continuity across each shell together with the usual jump condition for the radial derivative. Using a boundary integral approach based on the free Green kernel and single-layer potentials, we derive an explicit resolvent representation for an arbitrary number of shells with bounded coupling strengths. This yields a concrete Kre\u{\i}n-type formula and a boundary operator whose noninvertibility characterizes the discrete spectrum, and it is compatible with a 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 show 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. At the level of order-of-magnitude and qualitative trends, Type~I configurations yield a relatively strongly confined state, whereas Type~II configurations produce a comparatively shallow outer-shell state.