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Riesz \(L_p\) summability of spectral expansions related to the Schrödinger operator with constant magnetic field. (English) Zbl 1025.35012
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.

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35Q40 PDEs in connection with quantum mechanics
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
Full Text: DOI
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