Bound-like State in a 1D Self-Similar Delta-Barrier Array

We investigate a one-dimensional quantum system with a self-similar arrangement of delta-function potential barriers, exhibiting discrete scale invariance. The singular potential induces kinematically enforced symmetry breaking at $x=0$, decoupling the positive and negative spatial regions and leading to non-symmetric zero-energy states. We demonstrate that the system supports a unique zero-energy wavefunction, which, though not square-integrable, decays to zero at infinity and acts as a bound-like state with self-similar properties under discrete scaling transformations, akin to Efimov physics but limited to a single state. In momentum space, this wavefunction exhibits a threshold singularity at low momenta, with behavior depending on the scaling exponent $α$:power-law divergence and log-periodic modulations for $0 < α< 1$, logarithmic divergence for $α= 1$, and a finite limit for $α> 1$, which may be observable through time-of-flight or spectroscopic measurements in cold atom experiments. The system's continuous spectrum, starting at zero energy, lacks discrete bound states. These findings highlight the role of singular potentials in generating scale-invariant quantum phenomena and provide a minimal framework for studying discrete scale symmetry and its potential experimental signatures.