Asymptotic of the density of states for the Schrödinger operator with periodic electric potential
Let \(\Gamma\) be a lattice on \(\mathbb{R}^n\), \(V\) be a \(C^\infty\) \(\Gamma\)-periodic potential and \(P_V\) be the Schrödinger operator on \(L^2(\mathbb{R}^n)\), \(n\geq 2: P_V= \sum_{1\leq j\leq n} D^2_j+ V\). The authors prove that for the integrated density of states \(N(\mu, P_V)\), we have for any fixed \(\varepsilon> 0\), the following asymptotic expansion as \(\mu\to\infty\): \[ N(\mu, P_V)= a_n \mu^{n/2}+ a_{n-2} \mu^{(n- 2)/2}+ O(1)+ O(\mu^{(n- 3+\varepsilon)/2}) \] with \[ a_n= (2\pi)^{-n}|S^{n-1}|/n> a_{n- 2}= -{na_n\over 2} |K|^{-1} \int_K V(x) dx \] (\(K\) a periodic cell). They also recover a proof of the Béthe-Sommerfeld conjecture for \(n\leq 4\). The method is related to the one used by Hörmander for the study of the counting function of the eigenvalues of an elliptic operator and needs a periodic pseudodifferential calculus.
scientific article; zbMATH DE number 1425090
scientific article