Contact interactions, self-adjoint extensions, and low-energy scattering

Abstract

Low-energy scattering is well described by the effective-range expansion. In quantum mechanics, a tower of contact interactions can generate terms in this expansion after renormalization. Scattering parameters are also encoded in the self-adjoint extension of the Hamiltonian. We briefly review this well-known result for two particles with s-wave interactions using impenetrable self-adjoint extensions, including the case of harmonically trapped two-particle states. By contrast, the one-dimensional scattering problem is surprisingly intricate. We show that the families of self-adjoint extensions correspond to a coupled system of symmetric and antisymmetric outgoing waves, which is diagonalized by an SU(2) transformation that accounts for mixing and a relative phase. This is corroborated by an effective theory computation that includes all four energy-independent contact interactions. The equivalence of various one-dimensional contact interactions is discussed and scrutinized from the perspective of renormalization. As an application, the spectrum of a general point interaction with a harmonic trap is solved in one dimension.

arXiv comment31 pages, 1 figure

References in corpus (28)

• Feshbach Resonances in Ultracold Gases
• Modern Theory of Nuclear Forces
• Effective Field Theory for Few-Nucleon Systems
• A New Expansion for Nucleon-Nucleon Interactions
• Two-Nucleon Systems from Effective Field Theory
• Nuclear effective field theory: status and perspectives
• Effective Field Theory for Short-Range Forces
• Breathing Modes and Hidden Symmetry of Trapped Atoms in 2D
• Reduced Effective Lagrangians
• Self-consistent model of ultracold atomic collisions and Feshbach resonances in tight harmonic traps
• Rearranging Pionless Effective Field Theory
• How short is too short? Constraining zero-range interactions in nucleon-nucleon scattering
• Non-perturbative regularization and renormalization: simple examples from non-relativistic quantum mechanics
• The potential of effective field theory in NN scattering
• Causality and the effective range expansion
• Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics. I. General Theory
• Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter
• Four-Parameter Point-Interaction in 1-D Quantum Systems
• Causality and universality in low-energy quantum scattering
• From a Particle in a Box to the Uncertainty Relation in a Quantum Dot and to Reflecting Walls for Relativistic Fermions
• Self-adjoint Extensions for Confined Electrons:from a Particle in a Spherical Cavity to the Hydrogen Atom in a Sphere and on a Cone
• On Non-Relativistic Conformal Field Theory and Trapped Atoms: Virial Theorems and the State-Operator Correspondence in Three Dimensions
• A Riesz basis criterion for Schrödinger operators with boundary conditions dependent on the eigenvalue parameter
• The Solution of the Relativistic Schrodinger Equation for the ${\delta }^{\prime }$-Function Potential in 1-dimension Using Cutoff Regularization
• Removing the Wigner bound in non-perturbative effective field theory
• Supersymmetric Descendants of Self-Adjointly Extended Quantum Mechanical Hamiltonians
• Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for 1D delta-function potential
• Quantum anomaly and thermodynamics of one-dimensional fermions with antisymmetric two-body interactions