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 nonsymmetric zero-energy states. We demonstrate that the system supports a unique zero-energy wave function, which, although it is 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 wave function exhibits a threshold singularity at low momenta, with behavior depending on the scaling exponent $\alpha$: power-law divergence and log-periodic modulations for $0<\alpha <1$, logarithmic divergence for $\alpha =1$, and a finite limit for $\alpha >1$, 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.

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• Efimov states