Lifshitz tails for 2-dimensional random Schrödinger operators.
The authors prove that, at an open band edge, the density of states (DOS) of 2-dimensional continuous Anderson operators exhibits Lifshitz tails. Except for the dimension 1, the previous result was obtained by \textit{F. Klopp} [Duke Math. J. 98, No. 2, 335--396 (1999); correction ibid. 109, No. 2, 411--412 (2001; Zbl 1060.82509)] where it is proved that, taking \(0\) as the bottom of the spectrum, one has the Lifshitz behaviour \(\log| \log N(E)| \sim_{E\to 0^+} -\frac d2 \log E\) if and only if the DOS \(N_{\text{per}}\) of a well chosen underlying periodic Schrödinger operator satisfies \(\log N_{\text{per}}(E)\sim_{E\to 0^+} \frac d2 \log E\) (where \(d\) is the dimension). For 2-dimensional continuous Anderson models, the article under review answers the question of the existence of Lifshitz tails when the latter auxiliary property is not satisfied. As a first result, and under fairly general hypotheses on the Anderson potential and on the random variables, the authors show that \[ \lim_{E\to E_0}\frac{\log| \log | N(E) - N(E_0| | }{\log E} < 0, \] where \(E_0\) is the edge of the band. Under some extra hypotheses, and if \(E_0\) is simple, the Lifshitz exponent (given by the limit above) is shown to be determined by the behaviour of the Floquet eigenvalues of some background periodic Schrödinger operator. In particular, the Lifshitz exponent need not be \(\frac d2 = 1\).
scientific article; zbMATH DE number 1998318
scientific article