Erratum to: Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

Annales Henri Poincaré, Jul 2015

Jan Dereziński, Marcin Napiórkowski

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Erratum to: Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

Erratum to: Ann. Henri Poincare Erratum to: Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit Jan Derezinski 0 1 Marcin Napiorkowski 0 1 0 Current address: Marcin Napi orkowski Institute of Science and Technology Austria (IST Austria) Am Campus 1 3400 Klosterneuburg , Austria 1 Jan Derezin ski and Marcin Napi orkowski Department of Mathematical Methods in Physics Faculty of Physics, University of Warsaw Pasteura 5 02-093 Warsaw , Poland In the first section of the original article we stated that the interaction potential v is an even real function on Rd satisfying the following assumptions: where v denotes the Fourier transform given by v(p) = Rd v(x) eipx dx. Unfortunately, these assumptions seem not sufficient for the proof of the main result of the original article. However, all the arguments of that paper are correct if we replace the condition (4) by the condition p 2L Zd there exists C such that; for L 1 1. Conditions on Potentials v(p) C. Note that, even though (4) and (4 ) are closely related, neither of these conditions implies the other one. Some readers may complain that (4 ) looks somewhat complicated. Therefore, we give yet another condition, which looks easier and which implies (4 ): there exists C and > d such that |v(p)| C(1 + |p|). In fact, (2) implies that v is continuous and (4 ) implies that v L1. Then an easy argument involving Riemann sums and the Lebesgue Dominated Convergence Theorem yields Clearly, (4 ) is an immediate consequence of (1.1). 2. Periodization of Potentials One of the concepts used in the original article is the periodization of a potential. Below we would like to give a discussion of this concept which is somewhat more careful from the one contained in the original article. Suppose that v L1(Rd) and L > 0. Then the following formula vL(x) = nZd Then the Poisson summation formula shows that vL(x) = |v(p)| < . (See, e.g. [2, Thm. 2.4] or [1, Sect. 4.2.2].) In the original article we used (2.3) to define vL. Strictly speaking, this was not a mistake, at least under the conditions (1), (4 ) (which clearly imply (2.2) for all L), since then the definitions (2.3) and (2.1) are equivalent. However, one can argue that the definition (2.1) is more natural and slightly more general and thus we should have used it in the original article. In the original article we wrote that vL(x) v(x) as L . The meaning of that statement can be the following: if v L1(Rd) and I is a compact subset of Rd, then vL v in L1(I). In fact, let I [L0/2, L0/2[d. Then for L > L0 we have |v(x) vL(x)|dx nZd\{0} I |v(x + nL)|dx L 0. v(x) 0 vL(x) 0. We use this fact in the Proof of Lemma 4.1 of the original article. [1] Pinsky , M. : Introduction to Fourier Analysis and Wavelets . American Mathematical Society, Roanoke ( 2002 ) [2] Stein , E.M. , Weiss , G. : Introduction to Fourier Analysis on Euclidean Spaces . Princeton University Press, Princeton ( 1971 ) (...truncated)


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Jan Dereziński, Marcin Napiórkowski. Erratum to: Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit, Annales Henri Poincaré, 2015, pp. 1709-1711, Volume 16, Issue 7, DOI: 10.1007/s00023-014-0390-9