Background: Concerning the use of periodic boundary conditions for systems of $N$ (identical) particles, Ref. 1 considers a cube $\Lambda$ with volume $L^3$ and notes regarding the interaction potential of the form $V(x_1-x_2)$:
It is often useful, for calculations, to express $V$ in the plane-wave basis $|k\sigma\rangle$ with periodic boundary conditions. The Fourier transform of $V(x)$ in a cubic box $\Lambda$ is given by $$ \tilde V_L(k)=\int_\Lambda \mathrm dx\, e^{-ikx} \, V(x) \tag{3.111}$$ with corresponding Fourier series $$V_L(x) = \frac{1}{L^3} \sum\limits_k e^{ikx}\, \tilde V_L(k) \quad .\tag{3.112} $$ Note that $V_L(x)$ is a periodic function with period $L$ and is only equal to $V(x)$ if $x$ is in $\Lambda$. In the limit of infinite volume, however, $\tilde V_L(k)$ approaches the Fourier transform $\tilde V(k)$ of $V(x)$ in the whole space, $$ \lim\limits_{L\to\infty}\tilde V_L(k) = \int \mathrm d x\, e^{-ikx}\, V(x) = \tilde V(k) \tag{3.113}$$ and $$\lim\limits_{L\to\infty} V_L(x)=V(x) \quad .\tag{3.114} $$
The authors then proceed to express the two-body interaction in second quantization and again note that $V_L(x_1-x_2)$ is not equal to $V(x_1-x_2)$, only if $x_1-x_2 \in \Lambda$. But again they argue that in the limit of infinite volume, we recover the original potential. All of this more or less makes sense to me, but I have one doubt.
Question: It seems that it matters what cube exactly we take. Indeed, I see that if we take $\Lambda=[-L/2,L/2]^3 \subset \mathbb R^3$, then the limits in $(3.113)$ and $(3.114)$ yield the desired results. More generally, we can take some "sequence" of cubes which "converge" to $\mathbb R^3$, a notion which I think can be made rigorously.
However, by taking $\Lambda=[0,L]$ instead, I cannot see that this is true in general. First, it does not reproduce the Fourier transform in equation $(3.113)$. Moreover, for such an interval (I think for any asymmetric interval), the periodically extended interaction potential (cf. Ref. 4) $V_L$ is not symmetric anymore, which should be the case for identical particles.
But the authors do nowhere assume some special choice of $\Lambda$. Other References (see below) proceed and argue in the same manner.
References:
Many-Body Problems and Quantum Field Theory: An Introduction. Martin and Rothen. Section 3.2 page 111 and section 4 pages 135-137.
Nonequilibrium many-body theory of quantum systems. Stefanucci and Leuuwen. Appendix E, page 529.
Response Theory of the Electron-Phonon Coupling. Starke and Schober. Sections 2.2.1 and A.5. arxiv link.
Theoretical Solid State Physics In two Volumes. Volume 1. Albert Haug. Chapter II.B, section 24 (a), page 200.
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