# spectral eigenvalue staircase and quantum system

in a d-dimensional system of Quantum physics , does the Eigenvalue staircase

$N(E)= \sum_{E_{n}\le E} 1$ determine ALL the properties of Quantum System ??

for example, let us assume that the System is Chaotic and that Spectral eigenvalue staircase is given by $N(E)= <N (E)>+ \frac{1}{\pi}ArgZ(1/2+i \sqrt E )$

with $Z(s)= \prod_{n=0}^{\infty}\prod_{l_{p}}(1-l_{p}^{-s-n})$ the Gutzwiller zeta function

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I'd say doubtful - the Heisenberg ferromagnet and antiferromagnet have the same eigenvalue spectrum, but without correlation functions there's no way to tell them apart! – wsc Jun 11 '12 at 15:51