How to “normalize”(?) a vector after fft in MATLAB?

I want to change from real-space repr. to momentum-space repr. I have a Hamilton-operator (Anderson-model), and I calculated some kind of entropy of its eigenstates (this is working, I see what I want). Next I want to change to momentum repr. using fft, and now my eigenstates in momentum space are not normalized (?). E.g. if I calculate the sum of the eigenstates^2, it has to be 1, but it does not work.

I tried to sum the eigenstates^2 and normalized with them, but It does not work (I show the code without any failed trying).

  N=100; %dim of matrix
    Nx=15; %number of points
    %because of log scale
    xmin = -3.0;
    xmax =  3.0;
    dx = (xmax - xmin)/(Nx-1);
    x = zeros(1,Nx); %x axis pre
    ss = zeros(1,Nx); %entropy pre
    spp=zeros(1,Nx); %entropy in Fourier space pre
    eps=1.0e-6;
    
    for ix=1:Nx
        %log scale
        x(ix) = xmin + (ix-1)*dx;
        xx = 10.0^x(ix);
        
        average_s=0;
        average_spp=0;
        %anderson modell
        W=xx;
        r=rand(1,N)*W-(W/2);
        A=diag(ones(1,N-1),1)+diag(ones(1,N-1),-1)+diag(r);
        %diagonalization
        [V,D]=eig(A);
        %PROBLEM HERE:
        %Fourier transformation
        P=fft(V)/(sqrt(2*pi)*N);
        P=abs(P);
        for j=1:N
            four_sum=0; square_sum=0; entropy=0;
            four_sum_p=0; square_sum_p=0; entropyp=0;
            for i=1:N
                %Fou
                probp=(P(i,j)).^2;
                square_sum_p=square_sum_p+probp;
                if probp>eps
                    entropyp=entropyp-probp*log(probp);
                end;
                four_sum_p=four_sum_p+probp.^2;
                
                %Real
                prob=V(i,j).^2;
                square_sum=square_sum+prob;
                if prob>eps
                    entropy=entropy-prob*log(prob);
                end;
                four_sum=four_sum+prob.^2;
            end
            qp=square_sum_p.^2/(four_sum_p);
            average_spp=average_spp+entropyp-log(qp);
             
            q=square_sum.^2/(four_sum);
            average_s=average_s+entropy-log(q);
        end
        ss(ix)=average_s/N;
        spp(ix)=average_spp/N;
    end
    plot(x,ss,x,spp);

The structural entropy in real space (ss vector) has the correct form, but in momentum space (spp) after fft is not look like what I want, and it is not normalized.