We consider a periodic pseudo-differential operator on the real line, which is a lower-order perturbation of an elliptic operator with a homogeneous symbol and constant coefficients. It is proved that the density of states of such an operator admits a complete asymptotic expansion at large energies. A few first terms of this expansion are found in a closed form.
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Alexander V. Sobolev, Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 55–92DOI 10.4171/RMI/449