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Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator. (English) Zbl 1171.35092
The authors establish the high energy expansion of the integrated density of states for two-dimensional Schrödinger operators with smooth, periodic potential. After Floquet-Bloch decomposition, the problem reduces to prove such an asymptotics for the eigenvalues counting function of Schrödinger operators depending on the quasi-momentum. The method of proof is essentially based on a perturbation argument borrowed from L. Parnovski [Ann. Henri Poincaré 9, No. 3, 457–508 (2008; Zbl 1201.81054)].
As one can see in literature, in the last decade several authors were interested in this question in dimension \(d\) and gave partial results (see the references). In particular, the expansion holds true for \(d=1\) (see D. Schenk and M. A. Shubin, [Math. USSR, Sb. 56, 473–490 (1987); translation from Mat. Sb., N. Ser. 128(170), No. 4(12), 474–491 (1985; Zbl 0604.34015)]) and, assuming the existence of the expansion, its coefficients have been computed [see E. Korotyaev, A. Pushnitski, Bull. Lond. Math. Soc. 35, No. 6, 770–776 (2003; Zbl 1075.35542)]. This quite long history shows that this problem is really difficult and that it is a great achievement to solve it, even for \(d=2\). We strongly recommend the reading of the introduction where the strategy of the proof is sketched in a pedagogic way.
The proof is quite technical and difficult to follow. The notation \([h]\) and \(h\) for the “integer” and the “fractional” part of h with respect to the dual lattice can be mixed up, respectively, with the usual integer part and the set with unique element \(h\), which are actually used. In Remark 1.1, the statement (1.9) should be expected to be valid, since no complete proof is provided. Notice that these two last drawbacks could have been removed during the refereing process. More importantly, the proof gives information on the coefficients of the expansion but does not explain why many of them are zero, as pointed out by the authors themselves just before Remark 1.1.
The authors perform an important and difficult step in the understanding of the high energy behaviour of the integrated density of states. But this is not the end of the story. One should look for a more enlightening proof and try to treat more general situations (\(d>2\), non smooth potentials).

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
47A55 Perturbation theory of linear operators
47G30 Pseudodifferential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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References:
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