Closed formula to diagonalize discretized (perhaps randomized) Laplacians

Question

I was wondering whether there is a closed formula for the eigenvalues and eigenvectors of the discretized Laplacian in (edit) $[0,1]^n$ with a uniform grid, using what I imagine is a $2n+1$ stencil.

Furthermore, I was wondering whether there is some existing material on how to create random weighted sparse (graph) Laplacian matrices from a random spectrum, that is, matrices $L$ with $L\mathbf{1}=0$, $L=L^t$, positive diagonal, negative off-diagonal entries and non-negative random spectrum.

My thoughts on creating $\Lambda$ myself were

lambda = rand(n,1); lambda(1) = 0;

then create $\chi$ and orthogonalize it using stable Gram-Schmidt

chi = randn(n,n); chi(:,1) = ones(n,1); chi = gram_schmidt(chi);

finally

L=chi*lambda*(chi');

but it does not accomplish the sign conditions that I mentioned.

My interest is in having infinitesimal generators $Q$ for CTMC (edit: Continuous-Time Markov Chain) with number of states $n\gg 1000$ and available accurate diagonalization, so I thought the most studied case would be the discretized Laplacian. Any previous work on this?

Edit: thanks to lightxbulb in the comments, this works

function [Ltensor,FTtensor] = laplacian_matrix(nx,n)
  # pkg install -forge control, signals, communication
  # this function creates the periodic Laplacian matrix in the n-th hypercube,
  # as well as the matrix that diagonalizes it orthogonally, with size nx
  # h = 1/nx; I kind of not care about rescaling the Laplacian
  L = (diag(2*ones(nx,1)) - diag(ones(nx-1,1),1) - diag(ones(nx-1,1),-1));
  L(1,end) = -1; L(end,1) = -1;
  FT = dftmtx(nx);
  Ltensor = zeros(nx^n,nx^n); FTtensor = zeros(nx^n,nx^n);
  for i=1:n
    addL = 1; addFT = 1;
    for j=1:n
      if i==j
        addL = kron(addL,L);
      else
        addL = kron(addL,eye(nx));
      end
      addFT = kron(addFT,FT);
    end
    Ltensor = Ltensor + addL;
    FTtensor = FTtensor + addFT; % this is clearly wrong I should only do this once but it still diagonalizes and I just realized right now
  end
  FTtensor = FTtensor/(nx^n); % as I said this is wrong I will edit it later
end