Riesz \(L_p\) summability of spectral expansions related to the Schrödinger operator with constant magnetic field
The authors consider the Schrödinger operator with constant magnetic field \[ H=-\sum_{j=1}^d \left(\left(\partial_{x_j}+i{b_j\over 2}y_j\right)^2+ \left(\partial_{y_j}-i{b_j\over 2}x_j\right)^2\right) -\Delta_l \] in \(L^p({\mathbb R}^n), n=2d+l\), where \(\Delta_l\) is the Laplacian in \({\mathbb R}^l\). Denote by \(E_\lambda\) the spectral function of \(H\). The Riesz summation operator of index \(\beta\) is defined as \(S^\beta_\lambda = \int_0^\lambda (1-t/\lambda)^\beta dE_t\). It is proved that if \(\beta > \max (n|{1\over p}-{1\over 2}|-{1\over 2},0)\) and \(|{1\over p}-{1\over 2}|> {1\over {n+1}}\) then the operators \(S^\beta_\lambda\) are uniformly in \(\lambda\) bounded in \(L_p\) and for any \(f\in L_p({\mathbb R}^n), S^\beta_\lambda f\to f\) in \(L_p({\mathbb R}^n)\). Standard ingredients of the proof of Riesz summability (restriction estimate and kernel estimate) are also proved.
scientific article; zbMATH DE number 1974975