Absence of the confinement–induced Efimov effect: a direct proof in a specific geometry

We consider a system of three particles with identical mass interacting via short-range potentials, such that two of the particles are on parallel lines in a plane and the third one is on a line perpendicular to this plane. In this geometry, we prove that the corresponding Schrödinger operator only has a finite number of eigenvalues under physically reasonable assumptions on the decay of the interaction potentials. Our result disproves a recent prediction made in physics literature.