Three resonating fermions in flatland: proof of the super Efimov effect and the exact discrete spectrum asymptotics

We consider the system of three nonrelativistic spinless fermions in two dimensions, which interact through spherically-symmetric pair interactions. Recently, a claim has been made by Nishida et al for the existence of the so-called super Efimov effect by Nishida et al (2013 Phys. Rev. Lett. 110 235301). 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 a system indeed has an infinite number of bound levels. We also prove that , where N(E) is the number of bound states with the energy less than . The value of this limit is exactly equal to the value derived in Nishida et al using the renormalization group approach. Our proof resolves a recent controversy about the validity of results in Nishida et al (2013 Phys. Rev. Lett. 110 235301).

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