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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=- j=1 d x j +ib j 2y j 2 + y j -ib j 2x j 2 -Δ l

in L p ( n ),n=2d+l, where Δ l is the Laplacian in l . Denote by E λ the spectral function of H. The Riesz summation operator of index β is defined as S λ β = 0 λ (1-t/λ) β dE t . It is proved that if β>max(n|1 p-1 2|-1 2,0) and |1 p-1 2|>1 n+1 then the operators S λ β are uniformly in λ bounded in L p and for any fL p ( n ),S λ β ff in L p ( n ).

Standard ingredients of the proof of Riesz summability (restriction estimate and kernel estimate) are also proved.

MSC:
35P10Completeness of eigenfunctions, eigenfunction expansions for PD operators
35J10Schrödinger operator
35Q40PDEs in connection with quantum mechanics
47F05Partial differential operators