We investigate the discrete spectrum of finitely many point interactions for Neumann and Robin Laplacians on special unbounded and exterior $C^{1,1}$ domains in dimensions two and three. The operators are realized as self-adjoint extensions through an ordinary boundary triple whose gamma field and Weyl matrix are constructed from the Robin Green kernel. Eigenvalues below the background spectrum are characterized by the Weyl matrix, yielding an exact finite-dimensional counting formula. If the background operator is non-negative, this also gives the number of negative eigenvalues without assuming a finite zero-energy limit of the Weyl matrix. In the one-centre case we identify the critical coupling and prove that the unique eigenvalue branch is real analytic, strictly increasing, and strictly concave. For scalar multicentre couplings $\Theta=\alpha I_N$, sufficiently strong attraction produces exactly $N$ eigenvalues below the background spectrum, and all of them have universal leading asymptotics coinciding with the whole-space laws. If the bottom of the background spectrum is an isolated eigenvalue, the branch exists for every finite coupling and we determine its leading decoupling asymptotics as the coupling tends to $+\infty$. Explicit exterior-sphere and exterior-disk models illustrate the critical couplings and their threshold behaviour.
We study one-centre point interactions for the Dirichlet Laplacian on unbounded domains in dimensions two and three, with emphasis on exterior domains and special Lipschitz domains. These operators are singular perturbations constructed as self-adjoint extensions of the Dirichlet Laplacian restricted to functions vanishing at the interaction centre, and their resolvents are given by an explicit Kre\u{\i}n formula with a single extension parameter $\alpha$. The negative spectrum is completely characterized by a scalar equation and the critical coupling $\alpha$ separating binding from non-binding is the threshold limit of the Weyl function appearing in the Kre\u{\i}n formula. We establish domain monotonicity of the Weyl function, of the critical coupling, and of the unique negative eigenvalue when existing, and we derive sharp near-boundary asymptotics of the critical coupling in uniformly $C^{1,1}$ geometries. These estimates imply that, for every fixed coupling, nonpositive spectrum disappears when the interaction centre approaches the Dirichlet boundary. We also prove limiting absorption principles and purely absolutely continuous positive spectrum for a point-interaction in exterior domains case and in classes of special Lipschitz domains. We also analyze in depth several threshold phenomena. We show that the critical coupling is governed by the far-field behavior of the zero-energy Green function: exterior domains give threshold resonances, domains contained in a three-dimensional half-space give threshold eigenvalues, the half-plane gives a $p$-wave resonance, and planar wedges exhibit types of threshold states that are aperture-dependent. Finally, low-energy resolvent expansions are computed in the model cases and persistence or disappearance of eigenvalues at threshold are studied. The present paper seems to be the first systematic work on the subject.
D. Noja, Francesco Raso Stoia· 1 citation
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