In a dilute quantum gas at low temperature the interparticle distance is typically much larger than the effective range of the two-body interaction. In this physical regime, many properties are independent of the detailed form of the interaction potential and only depend on a few low-energy parameters, like the scattering length. Therefore, for the mathematical modelling of such systems it is convenient to replace the two-body interaction by an idealized $\delta$-interaction.
After a short introduction to ultracold quantum gases and many-body Hamiltonians with $\delta$-interactions, we consider a Bose gas interacting by $\delta$-interactions in one space dimension. We prove that the Hamiltonian of this system naturally arises as a resolvent limit $\epsilon\to 0$ of Schroedinger operators $H_{\epsilon}$ ($\epsilon > 0$), where the corresponding potentials scale like a Dirac sequence in $\epsilon > 0$.
Afterwards, we characterize the domain of the resulting Hamiltonian and we show that our result extends a previous result of Basti et al., concerning the three-body case, to the case of an arbitrary number of bosons $N \in\mathbb N$.
The talk is based on joint work with Marcel Griesemer and Ulrich Linden.
