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Schrodinger operators with concentric δ-shell interactions
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DOI：10.1007/s13324-026-01191-w.png)
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Abstract
En 中文
We study Schrodinger operators on R(3 )with finitely many concentric spherical delta-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 alpha(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 Krein-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.
Keywords:
delta-shell interaction
Singular perturbation
Resolvent formula
Krein-type formula
Partial-wave decomposition
Tunneling splitting
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IF:
1.6
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87
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