Universal Low-energy Behavior in a Quantum Lorentz Gas with Gross-Pitaevskii Potentials
We consider a quantum particle interacting with N obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form N2V (Nx) (Gross-Pitaevskii potential). We show convergence of the N dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for N large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.
KeywordsQuantum Lorentz gas Gross-Pitaevskii potentials Effective behavior Low-energy neutron scattering
Mathematics Subject Classification (2010)81Q99 81V35 82C10
The authors gratefully acknowledge the support from the GNFM Gruppo Nazionale per la Fisica Matematica - INDAM.
- 1.Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. 2nd edn. With an appendix by P. Exner. AMS Chelsea Publishing (2005)Google Scholar
- 13.Jeblick, M., Pickl, P.: Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials. arXiv:1801.04799(2018)
- 18.Papanicolaou, G.C., Varadhan, S.R.S.: Diffusion in regions with many small holes. In: Lecture notes in control and information sciences, vol. 25, pp 190–206. Springer, Berlin (1980)Google Scholar
- 20.Reed, M., Simon, B.: Methods of modern mathematical physics. Vol I: Functional Analysis. Academic Press, Cambridge (1981)Google Scholar