Statistics of addition spectra of independent quantum systems

Motivated by recent experiments on large quantum dots, we consider the energy spectrum in a system consisting of N particles distributed among K < N independent subsystems, such that the energy of each subsystem is a quadratic function of the number of particles residing on it. On a large scale, the ground-state energy E(N) of such a system grows quadratically with N, but in general there is no simple relation such as E(N) = aN + bN². The deviation of E(N) from exact quadratic behaviour implies that its second difference (the inverse compressibility) XN ≡ E(N + 1) - 2E(N) + E(N - 1) is a fluctuating quantity. Regarding the numbers XN as values assumed by a certain random variable X, we obtain a closed-form expression for its distribution F(X). Its main feature is that the corresponding density P(X) = dF(X)/dX has a maximum at the point X = 0. As K → ∞ the density is Poissonian, namely, P(X) → ^e-X.