Quantum Mechanics of Point Interactions

Quantum mechanics of point interactions concerns the theoretical and mathematical framework in which interactions between particles are idealised as zero-range, localised at a single point. This paradigm employs self-adjoint extensions of the free-particle Hamiltonian and distributional representations of singular potentials—most notably the Dirac delta and its derivatives—to define rigorous models of contact forces. Through renormalisation of coupling constants or the introduction of effective multi-body terms, one attains well-defined operators with bounded spectra. These constructions underpin fundamental understanding of scattering, bound states and resonance phenomena in one, two and three dimensions. Applications span quantum wires, ultracold atomic gases, nanostructures and mathematical toy models of quantum field theory. Recent advances have extended the formalism to curved spaces, few-body problems and high-dimensional geometries, highlighting the interplay between spectral theory, operator methods and renormalisation techniques with global implications for both theoretical physics and precision experiments.

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Technical terms

Point interaction: An idealised contact potential concentrated at a single point, typically represented by a delta function or its derivatives.

Self-adjoint extension: A mathematical procedure to define a quantum Hamiltonian operator on a domain where it is symmetric and its spectrum is real and complete.

Renormalisation: A method to remove infinities or singularities from a model by adjusting interaction parameters or introducing counterterms.

Delta potential: A potential proportional to the Dirac delta distribution, modelling an infinitely narrow and strong barrier or well.

Resolvent: The inverse of the operator (Hamiltonian minus a complex parameter), whose kernel encodes spectral and scattering information.

References

1. Distributional approach to point interactions in one-dimensional quantum mechanics. Frontiers in Physics (2014).
2. Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three. Annales Henri Poincaré (2022).
3. Point Potentials on Euclidean Space, Hyperbolic Space and Sphere in Any Dimension. Annales Henri Poincaré (2024).