The interacting Bose gas: A continuing challenge
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LENARY SESSION
The Interacting Bose Gas: A Continuing Challenge1, 2 Jakob Yngvason Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria DOI: 10.1134/S1063779610060110 21
1. INTRODUCTION
are trapped in an external potential V is
The experimental realization of Bose–Einstein condensation (BEC) in dilute, trapped alkali gases in 1995 [1, 2] has created lasting interest in the strange quantum properties exhibited by such systems. On the theoretical side the subject goes back to Einstein’s paper on BEC in ideal gases from 1924 [3], but the the ory of interacting Bose gases began with N.N. Bogoli ubov’s fundamental work of 1947 [4]. This was fol lowed by a period of considerable activity in this field in the late 1950’s and early 60’s, e.g., [5]. But mathe matically rigorous results were few and hard to get. In fact, after more than 60 years of theoretical investiga tion on interacting Bose gases, Mathematical Physics still faces the challenge to derive some of the funda mental properties of the low energy states of the many body Hamiltonian by rigorous mathematical analysis. Substantial progress, however, has been made in the past 10 years on the following topics: • The energy of a dilute Bose gas • Trapped Gases and the Gross–Pitaevskii equation • BEC and superfluidity in dilute, trapped gases • BEC and spontaneous symmetry breaking • Dimensional reduction in tightly confining traps • BEC and quantum phase transitions in optical lattices • Rotating gases and quantized vortices It is here only possible to discuss briefly a few of these topics. A general reference on the first six items from the Mathematical Physics point of view is [6] while the monograph [7] is devoted to rotating gases. See also [8–11] for some recent results on rotating gases and further references. 2. THE MATHEMATICAL SETTING The basic quantum mechanical Hamiltonian for N particles in ⺢3 that interact via a pair potential v and 1 Lecture
given at the International Bogolyubov Conference, Dubna, August 24, 2009. 2 The article is published in the original.
N
H =
∑ (– ∇
2 j
+ V ( xj ) ) +
j=1
∑
v ( x i – x j ).
(1)
1≤i
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