Schrödinger operators with concentric $δ$--shell interactions
Masahiro Kaminaga
TL;DR
This paper develops a self-contained boundary-integral framework for Schrödinger operators in $\mathbb{R}^3$ with finitely many concentric $\delta$-shell interactions, culminating in a Krein-type resolvent formula $R(z)=R_0(z)-\Gamma(z)\Theta K_N(z)^{-1}\Gamma(\bar z)^*$ where the discrete spectrum corresponds to noninvertibility of the boundary operator $K_N(z)$. Exploiting rotational symmetry, the authors reduce the problem to finite-dimensional blocks $m_\ell(z)$ and obtain explicit spectral conditions $\det(I+m_\ell(z)\Theta)=0$ in each angular momentum channel. In the two-shell, constant-coupling case they provide a detailed $s$-wave analysis: the negative spectrum is described by a closed secular equation $F_d(\kappa)=0$, and the ground state, if present, lies in the $s$-wave channel; they also count bound states for large shell separation and prove a tunneling splitting when the single-shell levels coincide. The results connect to core-shell quantum-dot physics (Type I/II configurations) and yield qualitative trends for confinement and tunneling, while the methodology combines boundary-integral techniques with partial-wave reductions to yield transparent, solvable spectral criteria.
Abstract
We study Schrödinger operators on $\mathbb R^3$ with finitely many concentric spherical $δ$--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 Kreĭn--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.