I will discuss a recent result about the excitation spectrum of a dilute Bose gas. We start from the microscopic many-body Hamiltonian H_N of quantum mechanics and derive Bogoliubov’s prediction for the first eigenvalues of H_N in the Gross—Pitaevskii regime for the inhomogeneous case (with an external field). We show that in this limit, the Hamiltonian H_N can be well approximated to the second order by a quadratic Hamiltonian describing a system of non-interacting quasi-particles. This is joint work with P.T. Nam
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