The Dirichlet Laplacian with a point interaction on unbounded Lipschitz domains

Abstract: 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 Kreĭn formula with a single extension parameter $α$. The negative spectrum is completely characterized by a scalar equation and the critical coupling $α$ separating binding from non-binding is the threshold limit of the Weyl function appearing in the Kreĭn 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.

• The paper establishes a complete spectral framework for the Dirichlet Laplacian with a point interaction using self-adjoint extension theory and explicit Krein resolvent formulas.
• It characterizes the negative spectrum and critical coupling with detailed asymptotic analyses near the Dirichlet boundary in both two and three dimensions.
• The study confirms the absolute continuity of the positive spectrum and elucidates threshold phenomena and domain monotonicity, providing insights for quantum and scattering applications.

Spectral Analysis of the Dirichlet Laplacian with Point Interactions on Unbounded Lipschitz Domains

Introduction and Motivation

This work establishes a comprehensive spectral theory for the Dirichlet Laplacian perturbed by a single point interaction on unbounded Lipschitz domains in two and three dimensions. Point interactions—singular perturbations localized at a single point—serve as tractable models for short-range interactions in both classical and quantum physical settings, particularly when the characteristic wavelength overwhelms the interaction's spatial extent. While self-adjoint extension theory for point interactions on $\mathbb R^d$ with $d \leq 3$ is classical, systematic treatment in unbounded subdomains is substantially less developed, especially in the presence of Dirichlet boundaries.

The focus is on two core geometric settings: exterior domains (the complement of a compact obstacle with Lipschitz boundary) and special Lipschitz domains (unbounded domains above a global Lipschitz graph). Both are quasi-conical, encompassing nontrivial boundary geometries and a spectrum of asymptotic behaviors at infinity. The analysis is restricted to a single interaction center, but the techniques developed provide a robust framework for extension.

Self-Adjoint Extension and Kre\u{\i}n Formula

The model operator is the Dirichlet Laplacian $A_D = -\Delta_D^\Omega$ on $L^2(\Omega)$, restricted to functions vanishing at $y \in \Omega$. The deficiency index analysis yields $(1,1)$, and the family of self-adjoint extensions $A_{\alpha, y}^\Omega$ is parametrized by a real coupling constant $\alpha$, including the unperturbed case $\alpha = \infty$.

Explicit Kre\u{\i}n resolvent formulas are derived, with the resolvent of $A_{\alpha, y}^\Omega$ expressed as a rank-one perturbation of the Dirichlet resolvent: $d \leq 3$0 where $d \leq 3$1 and $d \leq 3$2 is the Weyl function determined from the diagonal regular part of the Dirichlet Green function. The local singularity structure (Coulomb in $d \leq 3$3, logarithmic in $d \leq 3$4) is fully characterized, and the parameter $d \leq 3$5 encodes the boundary condition at the singularity.

Negative Spectrum and Critical Coupling

The spectrum below the essential threshold is exhaustively characterized by a scalar equation for $d \leq 3$6: $d \leq 3$7 In both dimensions, there is at most a single, simple negative eigenvalue. The threshold between binding and non-binding, $d \leq 3$8, is determined by the zero-energy limit of the Weyl function:

• In $d \leq 3$9, $A_D = -\Delta_D^\Omega$0 (locally determined by the boundary), while in $A_D = -\Delta_D^\Omega$1, $A_D = -\Delta_D^\Omega$2 requires cancellation of the free Green function's logarithmic divergence, resulting in a nontrivial renormalization.

Notably, as the interaction center $A_D = -\Delta_D^\Omega$3 approaches the Dirichlet boundary, $A_D = -\Delta_D^\Omega$4 diverges:

• In $A_D = -\Delta_D^\Omega$5: $A_D = -\Delta_D^\Omega$6
• In $A_D = -\Delta_D^\Omega$7: $A_D = -\Delta_D^\Omega$8

Crucially, for every fixed $A_D = -\Delta_D^\Omega$9, there exists a sufficiently small neighborhood near the boundary where the spectrum is purely nonnegative; the point interaction no longer produces a negative eigenvalue as $L^2(\Omega)$0.

Domain Monotonicity

The study establishes and exploits domain monotonicity:

• Expansion of the domain increases the critical coupling and lowers the (negative) eigenvalue, when it exists.
• This behavior generalizes the standard monotonicity for the Dirichlet Laplacian under inclusion and is essential for near-boundary asymptotics.

Positive Spectrum, Absolute Continuity, and Limiting Absorption

A limiting absorption principle is transferred from the background Dirichlet Laplacian to the point-interaction Laplacian via the Kre\u{\i}n formula. For both exterior and admissible special Lipschitz domains, the positive spectrum is purely absolutely continuous: $L^2(\Omega)$1 No embedded eigenvalues or singular continuous spectrum arise on the positive real axis.

Threshold Phenomena and Far-Field Influence

The behavior at the spectral threshold $L^2(\Omega)$2 depends critically on the far-field structure of the domain and the associated Green function:

• For three-dimensional exterior domains, a monopole-type resonance (decaying as $L^2(\Omega)$3 at infinity) appears at critical coupling.
• In three-dimensional domains contained in or asymptotic to a half-space, the critical state is $L^2(\Omega)$4 and gives rise to a threshold eigenvalue.
• In two dimensions, exterior domains exhibit $L^2(\Omega)$5-wave threshold resonances with constant far-field tails, while in the half-plane one finds $L^2(\Omega)$6-wave (dipole) resonances. Planar wedges offer a comprehensive catalog of angle-dependent threshold states with decay $L^2(\Omega)$7, $L^2(\Omega)$8 (where $L^2(\Omega)$9 is the wedge aperture).

The local geometry near the boundary controls divergence of the critical coupling, whereas the asymptotic geometry at infinity governs the threshold state type. These mechanisms are largely independent.

Model Geometries and Explicit Calculations

Extensive explicit calculations are provided:

• Half-space and half-plane: Analytical image methods yield exact Weyl functions and spectral characteristics.
• Exterior of balls and disks: Kelvin inversion and separation of variables produce complete Green function and spectrum descriptions, including near-boundary expansions.
• Planar wedges: Conformal mappings and Bessel function techniques quantify the threshold state's nature as a function of the aperture.

A summary table in the paper compiles critical coupling formulas for these canonical cases.

Low-Energy Expansions and Resonance Continuation

Low-energy resolvent expansions are derived in all cases. The sharp asymptotic form of the Kre\u{\i}n denominator determines whether approaching eigenvalue branches persist as resonances for $y \in \Omega$0 and the nature of these resonances. Specifically, in dimension two:

• For the half-plane ($y \in \Omega$1-wave resonance), the eigenvalue branch is persistent as a resonance through the threshold.
• For the exterior disk ($y \in \Omega$2-wave resonance), eigenvalues disappear at threshold and do not persist as resonances.

These findings parallel and extend recent results for regular potentials, with explicit connection to the spectral singularity structure induced by a singular perturbation.

Theoretical and Practical Implications

The results clarify fine distinctions in spectral and scattering theory for singular perturbations on unbounded domains, highlighting the additional complexity introduced compared to $y \in \Omega$3. From a mathematical physics standpoint, the work provides precise tools for analyzing zero-range interactions in realistic geometries, with direct application to wave mechanics, quantum graphs, and models of localized sources on open backgrounds. The explicit, quantitative nature of convergence rates near boundaries and the clear separation of local and far-field effects offer new guidance for both analytic study and numerical simulation.

Moving forward, generalization to multiple (finite or infinite) singularities, interactions on submanifolds, alternative boundary conditions, or the consideration of different domain topologies and noncompact manifolds are natural avenues. Also, understanding dynamical (time-dependent) manifestations of these spectral properties, such as resonance expansions for evolution equations, is invited by the present analysis.

Conclusion

By systematically developing the self-adjoint extension, resolvent structure, and spectral properties of the Dirichlet Laplacian with a single point interaction on broad classes of unbounded Lipschitz domains, this work provides foundational results for singular perturbations in complex geometric environments. The analysis encompasses explicit characterizations of negative eigenvalues, critical coupling asymptotics, domain monotonicity principles, absolute continuity on the positive spectrum, threshold resonance/eigenvalue classification, and low-energy scattering phenomena, establishing a robust platform for future exploration in both pure and applied settings.

Reference:

"The Dirichlet Laplacian with a point interaction on unbounded Lipschitz domains" (2607.04349)