Eigenvalues of the reference operator and semiclassical resonances

AbstractWe prove that the estimate of the number of the eigenvalues in intervals [λ−δ,λ+δ],0<hC⩽δ⩽C, of the reference operator L#(h) related to a self-adjoint operator L(h) is equivalent to the estimate of the integral over [λ−δ,λ+δ] of the sum of harmonic measures associated to the resonances of L(h) lying in a complex neighborhood Ω of λ>0 and the number of the positive eigenvalues of L(h) in [λ−δ,λ+δ]. We apply this result to obtain a Breit–Wigner approximation of the derivative of the spectral shift function near critical energy levels