Symmetry, Duality and Anholonomy of Point Interactions in One Dimension

Abstract

We analyze the spectral structure of the one dimensional quantum mechanical system with point interaction, which is known to be parametrized by the group U(2). Based on the classification of the interactions in terms of symmetries, we show, on a general ground, how the fermion-boson duality and the spectral anholonomy recently discovered can arise. A vital role is played by a hidden su(2) formed by a certain set of discrete transformations, which becomes a symmetry if the point interaction belongs to a distinguished U(1) subfamily in which all states are doubly degenerate. Within the U(1), there is a particular interaction which admits the interpretation of the system as a supersymmetric Witten model.

47 pages, 5 figures (with 7 EPS files); corrected typos

References in corpus (6)

• Fermion-Boson Duality of One-dimensional Quantum Particles with Generalized Contact Interaction
• Realizing Discontinuous Wave Functions with Renormalized Short-Range Potentials
• A Free Particle on a Circle with Point Interaction
• Double Spiral Energy Surface in One-dimensional Quantum Mechanics of Generalized Pointlike Potentials
• Duality and Anholonomy in Quantum Mechanics of 1D Contact Interactions
• A Remark on One-Dimensional Many-Body Problems with Point Interactions

Cited by in corpus (52)

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• Quasienergy anholonomy and its application to adiabatic quantum state manipulation
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• Cheon's anholonomies in Floquet operators
• Supersymmetric Quantum Mechanics with a Point Singularity
• Dynamical generation of fermion mass hierarchy in an extra dimension
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• Quantum Abacus
• Supersymmetric Quantum Mechanics under Point Singularities
• Majorana neutrinos with point interactions
• 6d Dirac fermion on a rectangle; scrutinizing boundary conditions, mode functions and spectrum
• Effects of Two Successive Parity-Invariant Point Interactions on One-Dimensional Quantum Transmission: Resonance Conditions for the Parameter Space
• A phenomenon of splitting resonant-tunneling one-point interactions
• Discrete Scale-Invariant Boson-Fermion Duality in One Dimension
• Resonance asymptotics in the generalized Winter model
• Non-Abelian Monopole in the Parameter Space of Point-like Interactions
• Inequivalent Quantizations of the N = 3 Calogero model with Scale and Mirror-S_3 Symmetry
• Supersymmetry in 6d Dirac Action
• Hierarchy of QM SUSYs on a Bounded Domain
• Resonant Transmission in One-Dimensional Quantum Mechanics with Two Independent Point Interactions: Full Parameter Analysis
• Role of a Phase Factor in the Boundary Condition of a One-Dimensional Junction
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• Wigner functions, contact interactions, and matching
• Boundary effect of a partition in a quantum well
• Path Integral on Star Graph
• Supersymmetric Descendants of Self-Adjointly Extended Quantum Mechanical Hamiltonians
• Continuous Spectra of Generalized Kronig-Penney Model
• Quantum Force Induced on a Partition Wall in a Harmonic Potential
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• Quantum Reflection and Transmission in Ring Systems with Double Y-Junctions: Occurrence of Perfect Reflection
• Quantum force due to distinct boundary conditions
• Origin of resonant tunneling through single-point barriers
• Quantum contact interactions
• Chaos in a one-dimensional integrable quantum system
• Quantum graph as a quantum spectral filter
• Peculiarity of Symmetric Ring Systems with Double Y-Junctions and the magnetic effects
• Supersymmetry and non-Abelian geometric phase for a free particle on a circle with point-like interactions