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)