Efimov-like Quasi-Bound States in a 1D Self-Similar Delta-Barrier Array

• URL: http://arxiv.org/abs/2505.01317v1
• Date: Fri, 02 May 2025 14:45:32 GMT
• Title: Efimov-like Quasi-Bound States in a 1D Self-Similar Delta-Barrier Array
• Authors: Jia-Chen Tang, Xu-Yang Hou, Yan He, Hao Guo,
• Abstract summary: We investigate a one-dimensional quantum system with a self-similar arrangement of potential barriers.<n>Our model features singular potentials with barriers positioned at geometrically scaled locations.<n>We show that the system supports a unique zero-energy wavefunction that is not square-integrable but decays to zero at infinity.
• Score: 2.8602214617572463
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate a one-dimensional quantum system with a self-similar arrangement of delta-function potential barriers, exhibiting discrete scale invariance. Unlike typical scale-invariant systems with smooth potentials, our model features singular potentials with barriers positioned at geometrically scaled locations, $x_n = x_0 \lambda^n$. We show that the system supports a unique zero-energy wavefunction that is not square-integrable but decays to zero at infinity, acting as a quasi-bound state. This wavefunction displays self-similarity under discrete scaling transformations, similar to the scaling symmetry in Efimov physics, though it represents a single state rather than a series of bound states. We analyze the wavefunction's asymptotic power-law decay and explore the spectral properties of the system, which lacks a discrete spectrum. These results highlight the role of singular potentials in generating scale-invariant quantum phenomena and provide a simple framework for studying discrete scale symmetry in quantum mechanics.

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