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We derive relations between various observables for N particles withzero-range or short-range interactions, in continuous space or on a lattice, intwo or three dimensions, in an arbitrary external potential. Some of ourresults generalise known relations between large-momentum behavior of themomentum distribution, short-distance behavior of the pair correlation functionand of the one-body density matrix, derivative of the energy with respect tothe scattering length or to time, and the norm of the regular part of thewavefunction; in the case of finite-range interactions, the interaction energyis also related to dE/da. The expression relating the energy to a functional ofthe momentum distribution is also generalised, and is found to break down forEfimov states with zero-range interactions, due to a subleading oscillatingtail in the momentum distribution. We also obtain new expressions for thederivative of the energy of a universal state with respect to the effectiverange, the derivative of the energy of an efimovian state with respect to thethree-body parameter, and the second order derivative of the energy withrespect to the inverse (or the logarithm in the two-dimensional case) of thescattering length. The latter is negative at fixed entropy. We use exactrelations to compute corrections to exactly solvable three-body problems andfind agreement with available numerics. For the unitary gas, we compare exactrelations to existing fixed-node Monte-Carlo data, and we test, with existingQuantum Monte Carlo results on different finite range models, our predictionthat the leading deviation of the critical temperature from its zero rangevalue is linear in the interaction effective range r_e with a model independentnumerical coefficient.

Exact relations for quantum-mechanical few-body and many-body problems with short-range interactions in two and three dimensions