Symmetry, Duality, and Anholonomy of Point Interactions in One Dimension

Abstract

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.

Supersymmetry in quantum mechanics with point interactions

“…A point interaction is specified by a characteristic matrix U ∈ U(2), and a wavefunction ϕ(x) and its derivative are required to obey the connection condition at, say x = 0 [10,8] (…”

“…The half-reflection transformation R is inherent in quantum mechanics with point singularities and is defined by 8) where Θ(x) is the Heaviside step function. The third transformation Q is defined by Q ≡ −iRP.…”

Classical aspects of quantum walls in one dimension

“…On the theoretical side, they have been found to exhibit a number of intriguing features, many of which have been seen before only in connection with quantum field theories. Examples include renormalization [1,2,3,4,5], Landau poles [6], anomalous symmetry breaking [5], duality [7,8,9], supersymmetry [9] and spectral anholonomy [9,10,11]. On the experimental side, the rapid developments of nanotechnology forecast that nano-scale quantum devices can be designed and manufactured into desired specifications.…”

An approximation to couplings on graphs

“…Such conditions exist, they were obtained independently in [FT00,CFT01] for a generalized point interaction, n = 2, and in [Ha00] for any n ≥ 1. It is easy to derive them: the self-adjointness requires vanishing of the boundary form, n j=1 (ψ j ψ ′ j −ψ ′ j ψ j )(0) = 0, which occurs iff the norms Ψ(0) ± iℓΨ ′ (0) C n with a fixed nonzero ℓ coincide, so the two vectors must be related by an n×n unitary matrix.…”