For a quantum mechanical system with a periodic Hamiltonian (SchrÃ¶dinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ that are less than or equal to $E$. (Or see the rigorous definition in Sec. 2 of Ref. 1.)

The Lifshitz exponent $\gamma_-$ is defined near the infimum $E_- = \inf \sigma(H)$ as (e.g. in Ref. 1):

$$\gamma_- = \lim_{E\downarrow E_-} \frac {\ln (-\ln N(E))}{\ln(E-E_-)},$$

and similarly near the supremum $E_+ = \sup \sigma(H)$, the analogous exponent can be defined as

$$\gamma_+ = \lim_{E\uparrow E_+} \frac {\ln (-\ln (1-N(E)))}{\ln(E_+-E)}.$$

These exponents measure how fast the density of eigenvalues grows or decays near the extreme ends of the spectrum.

I have some empirical data for a disordered quantum mechanical system for which I appear to measure $\gamma_- \ne \gamma_+$. This behavior does not seem to be observed in the model Hamiltonians that have been summarized in the review in Ref. 1 or in its references. I haven't found any literature that describes such behavior in any theoretical studies of quantum mechanical systems or random SchrÃ¶dinger operators.

As I am not an expert in this field, I would appreciate any pointers to references where such asymmetric behavior in the upper and lower tails of $N(E)$ have been observed.

Reference

- Kirsch and Metzger, http://arxiv.org/abs/math-ph/0608066