Proof of the Super Efimov Effect

We consider the system of 3 nonrelativistic spinless fermions in two dimensions, which interact through spherically-symmetric pair interactions. Recently a claim has been made for the existence of the so-called super Efimov effect [Y. Nishida et al., Phys. Rev. Lett. 110, 235301 (2013)]. Namely, if the interactions in the system are fine-tuned to a p-wave resonance, an infinite number of bound states appears, whose negative energies are scaled according to the double exponential law. We present the mathematical proof that such system indeed has an infinite number of bound levels. We also prove that $\lim_{E \to 0} |\ln|\ln E||^{-1} N(E) = 8/(3\pi)$, where $N(E)$ is the number of bound states with the energy less than $-E <0$. The value of this limit is equal exactly to the value derived in [Y. Nishida et al.] using renormalization group approach. Our proof resolves a recent controversy about the validity of results in [Y. Nishida et al.].