Does fft still work if I remove the zero?

Hi all, I am encountering some problems with regards to fft. I'm currently doing a physics project and it requires me to multiply a function say f(r) with a phase factor exp(1i.*V(r)) where V(r) is a potential which would go to infinity at r=0 and then lastly followed by a fft/ifft. I have no choice but to remove the 0 in my array of columns for the space (which is x and y as I later on define my r = sqrt(x.^2+y.^2)). I am wondering if I would still need the fftshift/ifftshift if my columns of x and y do not even have 0 in them? Or if I actually need the 0 for fft to work? I shall include my codes to make myself clearer..

 %space parameters, this is how I define my x and y, and r eventually
lower = -5;
upper = 5;
fq = 10;
x = (lower:1/fq:upper)';
y = (lower:1/fq:upper)'; 
L = length(x);
if mod(L,2)==1
    x((L+1)/2)=[];
    y((L+1)/2)=[];
    L = length(x);
    pr = fq*(1:(L/2))/(L/2)*pi;
    pl = -fliplr(pr);
    p = [pl pr];
end 
px = transpose(p); %columns
py = p; %rows
r = sqrt(x.^2+y'.^2); %x as columns y as rows
phi = atan2(y',x);
[xx,yy] = meshgrid(x,y);
[Px,Py] = meshgrid(px,py);
V = V = r.^2 + r.^(-2); 
 %this is a small section of my fft codes:
for t1= 0:dt:timef
     test = iw.*exp(-1i/6.*V.*dt); %position space
     test = fftshift(fft2(ifftshift(test)));%x->p
end